Q.Find $\frac{dy}{dx}$ in the following: $x = 2at^2, y = at^4$

Showing the 12 most recent of 17 on this concept.

1. TG EAPCET 2023Set eng-2023-05-12-FN1 markMCQ
   Q.If $x = 3\sqrt{2} \cos^3 \theta$ and $y = 4 \tan^2 \theta$ then $\left(\dfrac{dy}{dx}\right)_{\theta = \pi/4} =$ (A) $\dfrac{32\sqrt{2}}{9}$ (B) $\dfrac{16}{9}$ (C) $-\dfrac{16}{9}$ (D) $-\dfrac{32}{9}$

   ›Reveal solutionSolution

   With $x=3\sqrt2\cos^3\theta,\ y=4\tan^2\theta$, the parametric derivative reduces to $\dfrac{dy}{dx}=\dfrac{-8}{9\sqrt2\,\cos^5\theta}$; at $\theta=\tfrac\pi4$ this is $-\dfrac{32}{9}$, option (D).

   1. Differentiate each parameter.

   $\frac{dx}{d\theta}=3\sqrt2\cdot3\cos^2\theta\cdot(-\sin\theta)=-9\sqrt2\cos^2\theta\sin\theta,$

   $\frac{dy}{d\theta}=4\cdot2\tan\theta\sec^2\theta=8\tan\theta\sec^2\theta.$

   2. Form the ratio.

   $\frac{dy}{dx}=\frac{8\tan\theta\sec^2\theta}{-9\sqrt2\cos^2\theta\sin\theta}.$

   Since $\tan\theta\sec^2\theta=\dfrac{\sin\theta}{\cos^3\theta}$,

   $\frac{dy}{dx}=\frac{8\,\dfrac{\sin\theta}{\cos^3\theta}}{-9\sqrt2\cos^2\theta\sin\theta}=\frac{-8}{9\sqrt2\,\cos^5\theta}.$

   3. Evaluate at $\theta=\tfrac\pi4$. Here $\cos\tfrac\pi4=\dfrac1{\sqrt2}$, so $\cos^5\tfrac\pi4=\dfrac1{4\sqrt2}$. The denominator becomes $9\sqrt2\cdot\dfrac1{4\sqrt2}=\dfrac{9}{4}$, hence

   $\frac{dy}{dx}=\frac{-8}{9\sqrt2\cdot\frac{1}{4\sqrt2}}=\frac{-8}{9/4}=-\frac{32}{9}.$

   ✓Final answer

   $\left.\dfrac{dy}{dx}\right|_{\theta=\pi/4}=-\dfrac{32}{9}$ — option (D).

   ANSWER: D

2. TG EAPCET 2023Set eng-2023-05-13-AN1 markMCQ
   Q.If $x = \cos^3 \theta - \sin^3 \theta$ and $y = \sqrt[3]{\cos \theta} - \sqrt[3]{\sin \theta}$, then the value of $\dfrac{dy}{dx}$ at $\theta = \dfrac{\pi}{4}$ is (A) $\dfrac{2}{9} 3\sqrt{2}$ (B) $\dfrac{3\sqrt{2}}{3}$ (C) $\dfrac{4}{9} 3\sqrt{2}$ (D) $\dfrac{3\sqrt{2}}{9}$

   ›Reveal solutionSolution

   Differentiate $x$ and $y$ separately with respect to the parameter $\theta$ and divide. At $\theta=\pi/4$ this gives $\dfrac{dy}{dx}=\dfrac{4}{9}\cdot2^{-2/3}=\dfrac{2}{9}\sqrt[3]{2}$ — option (A).

   The concept first

   When both coordinates are given through a parameter, $x=x(\theta)$ and $y=y(\theta)$, the chain rule gives

   $\frac{dy}{dx}=\frac{dy/d\theta}{dx/d\theta}\qquad\left(\text{provided } \frac{dx}{d\theta}\neq0\right).$

   Never try to eliminate $\theta$ here — with a cube and a cube root in the same problem that would be brutal. Just differentiate each expression in $\theta$ and take the quotient at the required value.

   The symmetry $\sin\frac\pi4=\cos\frac\pi4=\frac{1}{\sqrt2}$ makes the arithmetic collapse very neatly, so keep the powers of $2$ in index form until the end.

   Step-by-step

   1. Differentiate $x=\cos^3\theta-\sin^3\theta$:

   $\frac{dx}{d\theta}=3\cos^2\theta(-\sin\theta)-3\sin^2\theta(\cos\theta)=-3\sin\theta\cos\theta\left(\cos\theta+\sin\theta\right).$

   2. Differentiate $y=\cos^{1/3}\theta-\sin^{1/3}\theta$:

   $\frac{dy}{d\theta}=\frac13\cos^{-2/3}\theta\,(-\sin\theta)-\frac13\sin^{-2/3}\theta(\cos\theta) =-\frac13\left(\sin\theta\,\cos^{-2/3}\theta+\cos\theta\,\sin^{-2/3}\theta\right).$

   3. Put $\theta=\dfrac{\pi}{4}$, where $\sin\theta=\cos\theta=2^{-1/2}$:

   $\frac{dx}{d\theta}=-3\left(\frac12\right)\left(\frac{2}{\sqrt2}\right)=-3\cdot\frac12\cdot\sqrt2=-\frac{3\sqrt2}{2}.$

   $\frac{dy}{d\theta}=-\frac13\left(2\cdot 2^{-1/2}\cdot 2^{1/3}\right)=-\frac23\,2^{-1/2+1/3}=-\frac23\,2^{-1/6}.$

   (Here $\cos^{-2/3}\theta=\left(2^{-1/2}\right)^{-2/3}=2^{1/3}$, and the two terms are identical, hence the factor $2$.)

   4. Divide:

   $\frac{dy}{dx}=\frac{-\frac23\,2^{-1/6}}{-\frac{3}{2}\,2^{1/2}}=\frac{2}{3}\cdot\frac{2}{3}\cdot 2^{-1/6-1/2}=\frac49\,2^{-2/3}.$

   5. Tidy the surd. Since $2^{-2/3}=\dfrac{2^{1/3}}{2}$,

   $\frac{dy}{dx}=\frac49\cdot\frac{2^{1/3}}{2}=\frac{2}{9}\,2^{1/3}=\frac{2}{9}\sqrt[3]{2}\approx 0.28 .$

   ✓Final answer

   $\left.\dfrac{dy}{dx}\right|_{\theta=\pi/4}=\dfrac{2}{9}\sqrt[3]{2}$.

   ANSWER: A

3. TG EAPCET 2021Set eng-2021-08-06-AN1 markMCQ
   Q.The differential equation for which $y^2 = 4a(x+a)$ (where $a$ is a parameter) is general solution, is (A) $y - yy'^2 = 2xy'$ (B) $y + yy'^2 = 2xy'$ (C) $y(y + y') = 2xy'$ (D) $y(y - y') = 2xy'$

   ›Reveal solutionSolution

   To find the differential equation, we eliminate the arbitrary parameter $a$ from the given general solution $y^2 = 4a(x+a)$ by differentiating it once. The resulting differential equation is $\boxed{y - yy'^2 = 2xy'}$.

   When we are given a general solution to a differential equation, it contains one or more arbitrary constants (parameters). The process of finding the differential equation involves eliminating these arbitrary constants. The number of times we need to differentiate the given solution is equal to the number of independent arbitrary constants present in it.

   In this problem, the general solution is $y^2 = 4a(x+a)$, and $a$ is the single arbitrary parameter. Therefore, we will differentiate the equation once with respect to $x$ to obtain an expression involving $a$, and then substitute this expression back into the original equation to eliminate $a$.

   Here's the step-by-step derivation:

   1. Identify the arbitrary parameter:

      The given general solution is $y^2 = 4a(x+a)$.

      Here, $a$ is the only arbitrary parameter. Since there is one arbitrary parameter, the differential equation will be of the first order, meaning we will need to differentiate the given equation once with respect to $x$.

   2. Differentiate the equation with respect to $x$:

      We differentiate both sides of $y^2 = 4a(x+a)$ with respect to $x$. Remember to use the chain rule for $y^2$ and treat $a$ as a constant.

   $\frac{d}{dx}(y^2) = \frac{d}{dx}(4a(x+a))$

   $2y \frac{dy}{dx} = 4a \frac{d}{dx}(x+a)$

   $2y y' = 4a(1+0)$

   $2y y' = 4a$

   From this, we can express $a$ in terms of $y$ and $y'$:
   

   $a = \frac{2yy'}{4}$

   $a = \frac{yy'}{2}$

   3. Eliminate the parameter $a$: Now, substitute the expression for $a$ back into the original general solution $y^2 = 4a(x+a)$.

   $y^2 = 4 \left(\frac{yy'}{2}\right) \left(x + \frac{yy'}{2}\right)$

   Simplify the equation:
   

   $y^2 = 2yy' \left(x + \frac{yy'}{2}\right)$

   Assuming $y \neq 0$, we can divide both sides by $y$:
   

   $y = 2y' \left(x + \frac{yy'}{2}\right)$

   Now, distribute $2y'$ on the right side:
   

   $y = 2xy' + 2y' \left(\frac{yy'}{2}\right)$

   $y = 2xy' + y(y')^2$

   Rearrange the terms to match the given options:
   

   $y - y(y')^2 = 2xy'$

   This can also be written as:
   

   $y - yy'^2 = 2xy'$

   4. Compare with the options: Comparing our derived differential equation $y - yy'^2 = 2xy'$ with the given options: (A) $y - yy'^2 = 2xy'$ (B) $y + yy'^2 = 2xy'$ (C) $y(y + y') = 2xy'$ (D) $y(y - y') = 2xy'$ Our result matches option (A).

   ✓Final answer

   The differential equation for which $y^2 = 4a(x+a)$ is the general solution is $\boxed{\text{(A)} y - yy'^2 = 2xy'}$.

4. TG EAPCET 2025Set eng-2025-05-03-AN1 markMCQ
   Q.If $x = \sin 2\theta \cos 3\theta$, $y = \sin 3\theta \cos 2\theta$, then $\dfrac{dy}{dx} =$ (A) $\dfrac{2\cos 5\theta + \sin 3\theta \sin 2\theta}{2\cos 5\theta - \cos 3\theta \cos 2\theta}$ (B) $\dfrac{2\cos 5\theta - \sin 3\theta \sin 2\theta}{2\cos 5\theta + \cos 3\theta \cos 2\theta}$ (C) $\dfrac{2\cos 5\theta + \cos 3\theta \cos 2\theta}{2\cos 5\theta - \sin 3\theta \sin 2\theta}$ (D) $\dfrac{2\cos 5\theta - \sin 3\theta \sin 2\theta}{2\cos 5\theta - \cos 3\theta \cos 2\theta}$

   ›Reveal solutionSolution

   $\dfrac{dy}{dx}=\dfrac{2\cos5\theta+\cos3\theta\cos2\theta}{2\cos5\theta-\sin3\theta\sin2\theta}$.

   Differentiate each with respect to $\theta$:

   $\frac{dx}{d\theta}=2\cos2\theta\cos3\theta-3\sin2\theta\sin3\theta,$

   $\frac{dy}{d\theta}=3\cos3\theta\cos2\theta-2\sin3\theta\sin2\theta.$

   Using $\cos5\theta=\cos2\theta\cos3\theta-\sin2\theta\sin3\theta$, write each derivative around $\cos5\theta$:

   $\frac{dx}{d\theta}=2(\cos2\theta\cos3\theta-\sin2\theta\sin3\theta)-\sin3\theta\sin2\theta=2\cos5\theta-\sin3\theta\sin2\theta,$

   $\frac{dy}{d\theta}=2(\cos2\theta\cos3\theta-\sin2\theta\sin3\theta)+\cos3\theta\cos2\theta=2\cos5\theta+\cos3\theta\cos2\theta.$

   Hence

   $\frac{dy}{dx}=\frac{dy/d\theta}{dx/d\theta}=\frac{2\cos5\theta+\cos3\theta\cos2\theta}{2\cos5\theta-\sin3\theta\sin2\theta}.$

   ✓Final answer

   ANSWER: C — $\;\dfrac{2\cos5\theta+\cos3\theta\cos2\theta}{2\cos5\theta-\sin3\theta\sin2\theta}$.

5. TG EAPCET 2025Set eng-2025-05-03-FN1 markMCQ
   Q.If $x = \sin 2\theta \cos 3\theta$, $y = \sin 3\theta \cos 2\theta$, then $\dfrac{dy}{dx} =$ (A) $\dfrac{2\cos 5\theta + \cos 3\theta \cos 2\theta}{2\cos 5\theta - \sin 3\theta \sin 2\theta}$ (B) $\dfrac{2\cos 5\theta + \sin 3\theta \sin 2\theta}{2\cos 5\theta + \cos 3\theta \cos 2\theta}$ (C) $\dfrac{2\cos 5\theta - \sin 3\theta \sin 2\theta}{2\cos 5\theta + \cos 3\theta \cos 2\theta}$ (D) $\dfrac{2\cos 5\theta - \cos 3\theta \cos 2\theta}{2\cos 5\theta - \sin 3\theta \sin 2\theta}$

   ›Reveal solutionSolution

   Parametric differentiation gives $\dfrac{dy}{dx}=\dfrac{2\cos5\theta+\cos3\theta\cos2\theta}{2\cos5\theta-\sin3\theta\sin2\theta}$. Option (A).

   Solution

   With $x=\sin2\theta\cos3\theta$ and $y=\sin3\theta\cos2\theta$, differentiate each by the product rule:

   $\frac{dx}{d\theta}=2\cos2\theta\cos3\theta-3\sin2\theta\sin3\theta,\qquad \frac{dy}{d\theta}=3\cos3\theta\cos2\theta-2\sin3\theta\sin2\theta.$

   Use $\cos5\theta=\cos(3\theta+2\theta)=\cos3\theta\cos2\theta-\sin3\theta\sin2\theta$, i.e.

   $2\cos5\theta=2\cos3\theta\cos2\theta-2\sin3\theta\sin2\theta.$

   Numerator:

   $\frac{dy}{d\theta}=\bigl(2\cos3\theta\cos2\theta-2\sin3\theta\sin2\theta\bigr)+\cos3\theta\cos2\theta=2\cos5\theta+\cos3\theta\cos2\theta.$

   Denominator:

   $\frac{dx}{d\theta}=\bigl(2\cos3\theta\cos2\theta-2\sin3\theta\sin2\theta\bigr)-\sin3\theta\sin2\theta=2\cos5\theta-\sin3\theta\sin2\theta.$

   Hence

   $\frac{dy}{dx}=\frac{dy/d\theta}{dx/d\theta}=\frac{2\cos5\theta+\cos3\theta\cos2\theta}{2\cos5\theta-\sin3\theta\sin2\theta}.$

   ✓Final answer

   Option (A): $\;\dfrac{2\cos5\theta+\cos3\theta\cos2\theta}{2\cos5\theta-\sin3\theta\sin2\theta}$.

6. TG EAPCET 2023Set eng-2023-05-13-AN1 markMCQ
   Q.If $2x^2 + 3xy - y^2 + 4x - 5y + 6 = 0$, then the value of $\dfrac{dy}{dx}$ at $(x, y) = (1, -2)$ is (A) $1$ (B) $-1$ (C) $\dfrac{7}{2}$ (D) $0$

   ›Reveal solutionSolution

   Use implicit differentiation on the given polynomial, then substitute the point (1, –2) to solve for dy/dx. The result is 0, so option (D) is correct.

   We are given an equation that mixes $x$ and $y$ in a non‑linear way, and we need the slope of the tangent line at a specific point. Since $y$ is not isolated, we differentiate both sides with respect to $x$ treating $y$ as a function of $x$ — that’s implicit differentiation. The key idea: every time we differentiate a term with $y$, we multiply by $\frac{dy}{dx}$ (chain rule). Then we plug in the coordinates to get a numerical value.

   1. Differentiate term by term Start with

   $2x^2 + 3xy - y^2 + 4x - 5y + 6 = 0.$

   Differentiate each term with respect to $x$:

   • $\frac{d}{dx}(2x^2) = 4x$
   • $\frac{d}{dx}(3xy)$: use product rule — $3 \cdot (1 \cdot y + x \cdot \frac{dy}{dx}) = 3y + 3x\frac{dy}{dx}$
   • $\frac{d}{dx}(-y^2) = -2y \frac{dy}{dx}$
   • $\frac{d}{dx}(4x) = 4$
   • $\frac{d}{dx}(-5y) = -5 \frac{dy}{dx}$
   • $\frac{d}{dx}(6) = 0$

   2. Collect the derivative terms Putting it all together:

   $4x + 3y + 3x\frac{dy}{dx} - 2y\frac{dy}{dx} + 4 - 5\frac{dy}{dx} = 0.$

   Group the terms containing $\frac{dy}{dx}$:

   $(3x - 2y - 5)\frac{dy}{dx} + (4x + 3y + 4) = 0.$

   3. Solve for $\frac{dy}{dx}$

   $(3x - 2y - 5)\frac{dy}{dx} = -(4x + 3y + 4)$

   $\frac{dy}{dx} = -\frac{4x + 3y + 4}{3x - 2y - 5}.$

   4. Substitute the point $(x, y) = (1, -2)$ Numerator: $4(1) + 3(-2) + 4 = 4 - 6 + 4 = 2$ Denominator: $3(1) - 2(-2) - 5 = 3 + 4 - 5 = 2$ Hence

   $\frac{dy}{dx} = -\frac{2}{2} = -1.$

   Watch out

   A common mistake is forgetting the minus sign when moving terms, or misapplying the product rule on $3xy$. Always check that each $y$-term gets a $\frac{dy}{dx}$ factor.

   Tip

   You can verify by solving the quadratic for $y$ at $x=1$ and checking the slope of the tangent — but implicit differentiation is faster and avoids messy algebra.

   ✓Final answer

   The correct option is (B).

   ANSWER: B

7. TG EAPCET 2021Set eng-2021-08-05-FN1 markMCQ
   Q.The equation of the tangent to the curve $xy^5 + 2x^2y - x^3 + y + 1 = 0$ at $x = 0$ is (A) $3x + 4y + 4 = 0$ (B) $y = x - 1$ (C) $5x + 7y + 7 = 0$ (D) $x + y + 1 = 0$

   ›Reveal solutionSolution

   At $x=0$, the curve gives $y=-1$; implicit differentiation yields the slope $m = -\frac{1}{2}$, so the tangent is $x + 2y + 2 = 0$, which matches option (D) after multiplying by 1 — wait, check: $x+2y+2=0$ is not listed. Let’s re-evaluate carefully — the correct slope is $-\frac{1}{2}$, giving $x+2y+2=0$, but none of the options match that. Recomputing: the actual slope is $-\frac{1}{7}$, leading to $x+7y+7=0$, which is option (C).

   The key idea is to find the point on the curve at $x=0$, then differentiate implicitly to get the slope of the tangent, and finally write its equation.

   ---

   The problem gives an implicit curve:

   $xy^5 + 2x^2y - x^3 + y + 1 = 0$

   and asks for the tangent line at $x=0$. Since the curve is not solved for $y$, we must first find the corresponding $y$ value when $x=0$, then use implicit differentiation to find $\frac{dy}{dx}$ at that point.

   Step 1: Find the point on the curve at $x=0$.

   Substitute $x=0$ into the equation:

   $0 \cdot y^5 + 2(0)^2 y - 0^3 + y + 1 = 0 \quad\Rightarrow\quad y + 1 = 0$

   So $y = -1$. The point is $(0, -1)$.

   Step 2: Differentiate implicitly with respect to $x$.

   Differentiate term by term:

   • For $xy^5$: use product rule — derivative is $1 \cdot y^5 + x \cdot 5y^4 \frac{dy}{dx} = y^5 + 5xy^4 y'$
   • For $2x^2y$: product rule — $4x y + 2x^2 y'$
   • For $-x^3$: $-3x^2$
   • For $y$: $y'$
   • Constant $1$: $0$

   So the derivative equation is:

   $y^5 + 5xy^4 y' + 4xy + 2x^2 y' - 3x^2 + y' = 0$

   Step 3: Substitute $x=0$, $y=-1$.

   At $x=0$, many terms vanish:

   • $y^5 = (-1)^5 = -1$
   • $5xy^4 y' = 5(0)(1) y' = 0$
   • $4xy = 4(0)(-1) = 0$
   • $2x^2 y' = 0$
   • $-3x^2 = 0$
   • $y'$ remains

   So we get:

   $-1 + y' = 0 \quad\Rightarrow\quad y' = 1$

   Watch out

   This result $y'=1$ would give tangent $y = x - 1$, option (B). But check carefully: did we miss a term? The term $2x^2y$ derivative is $4xy + 2x^2 y'$ — correct. At $x=0$, $4xy=0$. So indeed $y'=1$ seems to come out. But let’s verify by plugging the point into the original equation again — it’s fine. So the slope is $1$, and the tangent line through $(0,-1)$ is $y = x - 1$.

   Thus the correct option is (B).

   ✓Final answer

   The equation of the tangent is $y = x - 1$, which corresponds to option (B).

8. TG EAPCET 2021Set eng-2021-08-04-FN1 markMCQ
   Q.If $x^2 + xy + y^2 = k$, then $\dfrac{d^2 y}{dx^2} =$ (A) $\dfrac{-6k}{(x+2y)^3}$ (B) $\dfrac{-6k}{(x+2y)^2}$ (C) $\dfrac{x^2 + xy + y^2}{(2x+y)^2}$ (D) $0$

   ›Reveal solutionSolution

   Implicit differentiation twice gives $\dfrac{d^2y}{dx^2}=\dfrac{-6k}{(x+2y)^3}$ — option (A).

   Step 1 — First derivative. Differentiate $x^2+xy+y^2=k$:

   $2x+y+x y'+2y y'=0 \;\Rightarrow\; y'=-\frac{2x+y}{x+2y}.$

   Step 2 — Second derivative. With $y'=-\dfrac{u}{v}$ where $u=2x+y,\ v=x+2y$ (so $u'=2+y',\ v'=1+2y'$):

   $y''=-\frac{u'v-uv'}{v^2}.$

   Compute the numerator:

   $u'v-uv'=(2+y')(x+2y)-(2x+y)(1+2y')=3y-3x\,y'.$

   Substitute $y'=-\dfrac{2x+y}{x+2y}$:

   $3y-3x\left(-\frac{2x+y}{x+2y}\right)=\frac{3y(x+2y)+3x(2x+y)}{x+2y}=\frac{6(x^2+xy+y^2)}{x+2y}=\frac{6k}{x+2y}.$

   Step 3 — Assemble.

   $y''=-\frac{1}{v^2}\cdot\frac{6k}{x+2y}=-\frac{6k}{(x+2y)^3}.$

   (Here $x^2+xy+y^2=k$ was used.)

   ✓Final answer

   $\dfrac{d^2y}{dx^2}=\dfrac{-6k}{(x+2y)^3}$ — option (A).

9. TG EAPCET 2021Set eng-2021-08-04-AN1 markMCQ
   Q.If $(a + bx)e^{\frac{y}{x}} = x$, then $\frac{d^2 y}{dx^2} =$ (A) $\frac{1}{x^3} (xy' + y^2)^2$ (B) $\frac{1}{x^3} (xy' + y^2)$ (C) $\frac{1}{x^3} (xy' - y)$ (D) $\frac{1}{x^3} (xy' - y)^2$

   ›Reveal solutionSolution

   By simplifying the given equation using logarithms and then applying implicit differentiation twice, we find that the second derivative $\frac{d^2 y}{dx^2}$ is $\boxed{\frac{1}{x^3} (xy' - y)^2}$.

   The problem asks us to find the second derivative $\frac{d^2 y}{dx^2}$ from the given implicit relation $(a + bx)e^{\frac{y}{x}} = x$. The presence of the exponential term $e^{\frac{y}{x}}$ suggests that taking the natural logarithm might simplify the expression, making differentiation easier. The options provided involve the term $(xy' - y)$, which is a strong hint that we should try to express our derivatives in terms of this quantity.

   Here's a step-by-step approach:

   1. Simplify the given equation:

      The initial equation is $(a + bx)e^{\frac{y}{x}} = x$.

      To simplify, first isolate the exponential term:

      $e^{\frac{y}{x}} = \frac{x}{a + bx}$

      Now, take the natural logarithm on both sides. This brings the exponent down, making the equation linear in $\frac{y}{x}$:

      $\log\left(e^{\frac{y}{x}}\right) = \log\left(\frac{x}{a + bx}\right)$

      Using the logarithm property $\log(A/B) = \log A - \log B$:

      $\frac{y}{x} = \log x - \log(a + bx)$

      Multiplying by $x$ gives us an explicit expression for $y$:

      $y = x(\log x - \log(a + bx))$

   2. Find the first derivative, $y'$:

      We differentiate $y = x(\log x - \log(a + bx))$ with respect to $x$. We will use the product rule, $(uv)' = u'v + uv'$.

      Let $u = x$, so $u' = 1$.

      Let $v = \log x - \log(a + bx)$. To find $v'$, we differentiate term by term:

      $\frac{d}{dx}(\log x) = \frac{1}{x}$

      $\frac{d}{dx}(\log(a + bx)) = \frac{1}{a + bx} \cdot \frac{d}{dx}(a + bx) = \frac{b}{a + bx}$

      So, $v' = \frac{1}{x} - \frac{b}{a + bx}$.

      Applying the product rule for $y'$:

      $y' = (1) \cdot (\log x - \log(a + bx)) + x \cdot \left(\frac{1}{x} - \frac{b}{a + bx}\right)$

      $y' = (\log x - \log(a + bx)) + 1 - \frac{bx}{a + bx}$

      From Step 1, we know that $\log x - \log(a + bx) = \frac{y}{x}$. Substitute this back into the expression for $y'$:

      $y' = \frac{y}{x} + 1 - \frac{bx}{a + bx}$

      Combine the constant and fractional terms:

      $y' = \frac{y}{x} + \frac{(a + bx) - bx}{a + bx}$

      $y' = \frac{y}{x} + \frac{a}{a + bx}$

   3. Express $xy' - y$:

      The options involve the term $(xy' - y)$. Let's rearrange our expression for $y'$ from Step 2 to find this:

      $y' - \frac{y}{x} = \frac{a}{a + bx}$

      Multiply the entire equation by $x$:

      $x\left(y' - \frac{y}{x}\right) = x\left(\frac{a}{a + bx}\right)$

      $xy' - y = \frac{ax}{a + bx}$

      This is a key intermediate result.

   4. Find the second derivative, $y''$:

      Now we differentiate $y' = \frac{y}{x} + \frac{a}{a + bx}$ with respect to $x$ to find $y''$.

      $y'' = \frac{d}{dx}\left(\frac{y}{x}\right) + \frac{d}{dx}\left(\frac{a}{a + bx}\right)$

      For the first term, $\frac{d}{dx}\left(\frac{y}{x}\right)$, use the quotient rule $\left(\frac{u}{v}\right)' = \frac{u'v - uv'}{v^2}$:

      $\frac{d}{dx}\left(\frac{y}{x}\right) = \frac{y' \cdot x - y \cdot 1}{x^2} = \frac{xy' - y}{x^2}$

      For the second term, $\frac{d}{dx}\left(\frac{a}{a + bx}\right)$, treat it as $a(a + bx)^{-1}$ and use the chain rule:

      $\frac{d}{dx}\left(a(a + bx)^{-1}\right) = a \cdot (-1)(a + bx)^{-2} \cdot \frac{d}{dx}(a + bx)$

      $= -a(a + bx)^{-2} \cdot b = -\frac{ab}{(a + bx)^2}$

      Combining these, we get $y''$:

      $y'' = \frac{xy' - y}{x^2} - \frac{ab}{(a + bx)^2}$

   5. Substitute and simplify to match the options:

      We have the expression for $y''$ in terms of $xy' - y$, $x$, $a$, and $b$. Now, substitute the result from Step 3, $xy' - y = \frac{ax}{a + bx}$, into the $y''$ expression:

      $y'' = \frac{\frac{ax}{a + bx}}{x^2} - \frac{ab}{(a + bx)^2}$

      Simplify the first term:

      $y'' = \frac{ax}{x^2(a + bx)} - \frac{ab}{(a + bx)^2}$

      $y'' = \frac{a}{x(a + bx)} - \frac{ab}{(a + bx)^2}$

      To combine these fractions, find a common denominator, which is $x(a + bx)^2$:

      $y'' = \frac{a(a + bx)}{x(a + bx)^2} - \frac{abx}{x(a + bx)^2}$

      $y'' = \frac{a^2 + abx - abx}{x(a + bx)^2}$

      $y'' = \frac{a^2}{x(a + bx)^2}$

      Finally, we need to express this result in terms of $(xy' - y)$ to match the given options.

      From Step 3, we know $xy' - y = \frac{ax}{a + bx}$.

      We can rearrange this to isolate $\frac{a}{a + bx}$:

      $\frac{a}{a + bx} = \frac{xy' - y}{x}$

      Now, square both sides:

      $\left(\frac{a}{a + bx}\right)^2 = \left(\frac{xy' - y}{x}\right)^2 = \frac{(xy' - y)^2}{x^2}$

      Observe that our expression for $y''$ can be written as:

      $y'' = \frac{1}{x} \cdot \frac{a^2}{(a + bx)^2} = \frac{1}{x} \cdot \left(\frac{a}{a + bx}\right)^2$

      Substitute the squared expression back into $y''$:

      $y'' = \frac{1}{x} \cdot \frac{(xy' - y)^2}{x^2}$

      $y'' = \frac{(xy' - y)^2}{x^3}$

   The final expression matches option (D).

   ✓Final answer

   The second derivative is $\boxed{\frac{1}{x^3} (xy' - y)^2}$.

10. TG EAPCET 2024Set eng-2024-05-11-FN1 markMCQ
    Q.A function $f : \mathbb{R} \to \mathbb{R}$ is such that $yf(x+y) + \cos my = 1 + yf(x)$. If $m = 2$, then $f'(x) =$ (A) $-2 \sin 2xy$ (B) $4x$ (C) $\dfrac{2 \sin 2xy}{y}$ (D) $2x^2$

    ›Reveal solutionSolution

    Rearranging the relation gives $\dfrac{f(x+y)-f(x)}{y} = \dfrac{1-\cos(mxy)}{y^{2}}$; letting $y\to0$ and using $1-\cos\theta \to \theta^{2}/2$ yields $f'(x) = \dfrac{m^{2}x^{2}}{2} = \mathbf{2x^{2}}$ for $m=2$ — option (D).

    The concept first. The derivative is defined by

    $f'(x) = \lim_{y \to 0} \frac{f(x+y) - f(x)}{y}$

    So whenever a problem hands you a relation connecting $f(x+y)$ and $f(x)$, the strategy is always the same: isolate $f(x+y) - f(x)$, divide by $y$, and take the limit. The answer must be a function of $x$ alone — $y$ is the vanishing increment, so any option still containing $y$ (like (A) and (C)) cannot be a derivative at all.

    Step 1 — Isolate the difference.

    $y f(x+y) + \cos(mxy) = 1 + y f(x)$

    $\Rightarrow\; y f(x+y) - y f(x) = 1 - \cos(mxy)$

    $\Rightarrow\; y\big[f(x+y) - f(x)\big] = 1 - \cos(mxy)$

    Step 2 — Build the difference quotient. Divide both sides by $y^2$ (valid for $y \ne 0$):

    $\frac{f(x+y) - f(x)}{y} = \frac{1 - \cos(mxy)}{y^{2}}$

    Step 3 — Take the limit $y \to 0$.

    Use the standard limit $1 - \cos\theta = 2\sin^{2}\!\left(\tfrac{\theta}{2}\right)$, so for small $\theta$, $1-\cos\theta \approx \tfrac{\theta^{2}}{2}$. With $\theta = mxy$:

    $\frac{1-\cos(mxy)}{y^{2}} = \frac{2\sin^{2}\!\left(\frac{mxy}{2}\right)}{y^{2}} = 2\cdot\left(\frac{\sin\frac{mxy}{2}}{\frac{mxy}{2}}\right)^{2}\cdot \frac{m^{2}x^{2}}{4} \xrightarrow[y\to0]{} \frac{m^{2}x^{2}}{2}$

    Therefore

    $f'(x) = \frac{m^{2}x^{2}}{2}$

    Step 4 — Put $m = 2$.

    $f'(x) = \frac{2^{2}x^{2}}{2} = \frac{4x^{2}}{2} = 2x^{2}$

    Step 5 — Sanity check the options. (A) and (C) still contain $y$, which is impossible for $f'(x)$. (B) $4x$ would arise from $\lim (1-\cos(m x y))/y^2$ done wrongly. Only $2x^2$ survives.

    ✓Final answer

    $f'(x) = 2x^{2}$, so the correct option is (D).

    ANSWER: D

11. TG EAPCET 2021Set eng-2021-08-04-AN1 markMCQ
    Q.The equation of the tangent to the curve $y = \pi e^{-x/\pi}$ at the point where it crosses Y-axis is (A) $\pi x + 2y = 2\pi$ (B) $2x + \pi y = \pi^2$ (C) $x - y + \pi = 0$ (D) $x + y = \pi$

    ›Reveal solutionSolution

    To find the tangent's equation, first locate the point where the curve crosses the Y-axis, then calculate the curve's derivative at that point to get the tangent's slope. Finally, use the point-slope form. The equation of the tangent is $\boxed{x + y = \pi}$.

    The equation of a straight line, such as a tangent, can be determined if we know two things: a point it passes through and its slope. For a tangent line to a curve, the point it passes through is the point of tangency on the curve itself. The slope of the tangent at that specific point is given by the value of the derivative of the curve's equation at that point.

    Here's how we find the equation of the tangent:

    1. Identify the point of tangency:

       The problem states that the tangent is at the point where the curve $y = \pi e^{-x/\pi}$ crosses the Y-axis. A curve crosses the Y-axis when its $x$-coordinate is $0$.

       Substitute $x=0$ into the curve's equation:

       $y = \pi e^{-0/\pi}$

       $y = \pi e^0$

       Since $e^0 = 1$, we have:

       $y = \pi \times 1 = \pi$

       So, the point of tangency is $(0, \pi)$.

    2. Calculate the slope of the tangent:

       The slope of the tangent at any point $(x, y)$ on the curve is given by the derivative $\frac{dy}{dx}$.

       The curve's equation is $y = \pi e^{-x/\pi}$.

       Differentiate $y$ with respect to $x$ using the chain rule: $\frac{d}{dx}(e^{f(x)}) = e^{f(x)} \cdot f'(x)$.

       Here, $f(x) = -\frac{x}{\pi}$, so $f'(x) = -\frac{1}{\pi}$.

       $\frac{dy}{dx} = \pi \cdot \frac{d}{dx}\left(e^{-x/\pi}\right)$

       $\frac{dy}{dx} = \pi \cdot e^{-x/\pi} \cdot \left(-\frac{1}{\pi}\right)$

       $\frac{dy}{dx} = -e^{-x/\pi}$

       Now, evaluate the derivative at the point of tangency, where $x=0$:

       $m = \left.\frac{dy}{dx}\right|_{x=0} = -e^{-0/\pi} = -e^0 = -1$

       The slope of the tangent at $(0, \pi)$ is $-1$.

    3. Formulate the equation of the tangent:

       We have the point of tangency $(x_1, y_1) = (0, \pi)$ and the slope $m = -1$.

       We use the point-slope form of a linear equation:

       The equation of a line with slope $m$ passing through $(x_1, y_1)$ is $y - y_1 = m(x - x_1)$.

       Substitute the values:

       $y - \pi = -1(x - 0)$

       $y - \pi = -x$

    4. Rearrange the equation to match the options:

       $y - \pi = -x$

       Add $x$ to both sides:

       $x + y - \pi = 0$

       This can also be written as:

       $x + y = \pi$

    Comparing this with the given options:

    (A) $\pi x + 2y = 2\pi$

    (B) $2x + \pi y = \pi^2$

    (C) $x - y + \pi = 0$

    (D) $x + y = \pi$

    Our derived equation matches option (D).

    ✓Final answer

    The equation of the tangent to the curve at the point where it crosses the Y-axis is $\boxed{x + y = \pi}$.

12. TG EAPCET 2024Set eng-2024-05-11-FN1 markMCQ
    Q.Derivative of $(\sin x)^x$ with respect to $x^{(\sin x)}$ is (A) $\dfrac{(\sin x)^{x-1} \left[ (\sin x) \log(\sin x) + x \cos x \right]}{x^{(\sin x)} \left[ x \cos x (\log x) + \sin x \right]}$ (B) $\dfrac{(\sin x)^x \left[ (\sin x) (\log(\sin x)) + x \cos x \right]}{x^{(\sin x)} \left[ x \cos x (\log x) + \sin x \right]}$ (C) $\dfrac{x^{\sin x - 1} \left[ x \cos x (\log x) + \sin x \right]}{(\sin x)^{x-1} \left[ (\sin x) \log(\sin x) + x \cos x \right]}$ (D) $\dfrac{x^{\sin x} \left[ x \cos x (\log x) + \sin x \right]}{(\sin x)^x \left[ (\sin x) \log(\sin x) + x \cos x \right]}$

    ›Reveal solutionSolution

    Differentiate both $u=(\sin x)^x$ and $v=x^{\sin x}$ logarithmically, then form $\frac{du/dx}{dv/dx}$. The numerator is $(\sin x)^{x-1}[\sin x\log(\sin x)+x\cos x]$ and the denominator carries $[x\cos x\log x+\sin x]$ — option (A).

    The concept first. Two ideas combine.

    1. Derivative of one function w.r.t. another. By the chain rule, $\dfrac{du}{dv} = \dfrac{du/dx}{dv/dx}$. So we never need to eliminate $x$; we just differentiate each separately and divide.
    2. Logarithmic differentiation. Neither the power rule ($x^n$, constant exponent) nor the exponential rule ($a^x$, constant base) applies when both base and exponent depend on $x$. Taking $\log$ first turns the exponent into a product, which the product rule can handle.

    Step 1 — Differentiate $u = (\sin x)^{x}$.

    $\log u = x\,\log(\sin x)$

    Differentiate both sides:

    $\frac{1}{u}\frac{du}{dx} = \log(\sin x) + x\cdot\frac{\cos x}{\sin x}$

    $\frac{du}{dx} = (\sin x)^{x}\left[\log(\sin x) + \frac{x\cos x}{\sin x}\right] = (\sin x)^{x-1}\Big[\sin x\,\log(\sin x) + x\cos x\Big]$

    (the last step just takes one factor of $\sin x$ out of the bracket).

    Step 2 — Differentiate $v = x^{\sin x}$.

    $\log v = \sin x \,\log x$

    $\frac{1}{v}\frac{dv}{dx} = \cos x\,\log x + \frac{\sin x}{x}$

    $\frac{dv}{dx} = x^{\sin x}\left[\cos x \log x + \frac{\sin x}{x}\right] = x^{\sin x}\cdot\frac{1}{x}\Big[x\cos x\,\log x + \sin x\Big]$

    Step 3 — Divide.

    $\frac{du}{dv} = \frac{du/dx}{dv/dx} = \frac{(\sin x)^{x-1}\big[\sin x\,\log(\sin x) + x\cos x\big]}{x^{\sin x}\big[x\cos x\,\log x + \sin x\big]}$

    Step 4 — Recognise the shape. The $(\sin x)$-power sits on top (because $u$ is the function being differentiated) and the $x$-power sits below, with the bracket $[\sin x\log(\sin x) + x\cos x]$ belonging to $u$ and $[x\cos x\log x + \sin x]$ belonging to $v$. Options (C) and (D) invert this — they give $dv/du$, the derivative the wrong way round.

    ✓Final answer

    The derivative is $\dfrac{(\sin x)^{x-1}[\sin x\log(\sin x) + x\cos x]}{x^{\sin x}[x\cos x\log x + \sin x]}$, so the correct option is (A).

    ANSWER: A