Q.If $(x^2 + y^2)^2 = xy$, find $\dfrac{dy}{dx}$. OR If $x = a(2\theta - \sin 2\theta)$ and $y = a(1 - \cos 2\theta)$, find $\dfrac{dy}{dx}$ when $\theta = \dfrac{\pi}{3}$.

Showing the 12 most recent of 17 on this concept.

1. TG EAPCET 2026Set eng-2026-05-10-FN1 markMCQ
   Q.The differential equation corresponding to the family of curves $y = \log_e (ax + 3)$, where $a$ is an arbitrary constant is (A) $x \dfrac{dy}{dx} + 3e^{-x} = 1$ (B) $x \dfrac{dy}{dx} + 3e^{y} = 1$ (C) $x \dfrac{dy}{dx} + 3e^{-y} = 1$ (D) $x \dfrac{dy}{dx} + 3e^{x} = 1$

   ›Reveal solutionSolution

   The key idea is to eliminate the arbitrary constant $a$ by differentiating the given family and then substituting back. The correct differential equation is $x \frac{dy}{dx} + 3e^{-y} = 1$, which corresponds to option (C).

   We are given a family of curves $y = \log_e (ax + 3)$, where $a$ is an arbitrary constant. To find its differential equation, we need an equation involving $x$, $y$, and $\frac{dy}{dx}$ that holds for every curve in the family — meaning $a$ must be eliminated.

   The natural approach: differentiate the given relation, then use the original equation to replace $a$ in terms of $x$ and $y$.

   1. Differentiate both sides with respect to $x$. Since $y = \ln(ax + 3)$, we have

   $\frac{dy}{dx} = \frac{a}{ax + 3}.$

   2. Express $a$ from the original equation. From $y = \ln(ax + 3)$, exponentiate:

   $e^{y} = ax + 3 \quad \Rightarrow \quad ax = e^{y} - 3 \quad \Rightarrow \quad a = \frac{e^{y} - 3}{x}.$

   3. Substitute $a$ into the derivative. Replace $a$ in $\frac{dy}{dx} = \frac{a}{ax + 3}$:

   $\frac{dy}{dx} = \frac{\frac{e^{y} - 3}{x}}{e^{y}} = \frac{e^{y} - 3}{x e^{y}}.$

   4. Rearrange to match the given options. Multiply both sides by $x e^{y}$:

   $x e^{y} \frac{dy}{dx} = e^{y} - 3.$

   Bring terms together:

   $x e^{y} \frac{dy}{dx} - e^{y} = -3.$

   Factor $e^{y}$:

   $e^{y} \left( x \frac{dy}{dx} - 1 \right) = -3.$ …

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2. CA Foundation 2026Set may-20261 markMCQ
   Q.If $x^y = y^x$, then $\dfrac{dy}{dx} =$ ______. (A) $\dfrac{y(x \log y - y)}{x(y \log x - x)}$ (B) $\dfrac{y(x \log y + y)}{x(y \log x - x)}$ (C) $\dfrac{y(x \log y - y)}{x(y \log x + x)}$ (D) $\dfrac{x(x \log y - y)}{y(y \log x - x)}$

   ›Reveal solutionSolution

   Log-differentiate $x^y=y^x$: from $y\log x=x\log y$ you get $\dfrac{dy}{dx}=\dfrac{y(x\log y-y)}{x(y\log x-x)}$.

   Step 1 — Take logarithms of both sides

   $x^y=y^x\ \Rightarrow\ y\log x=x\log y$

   Step 2 — Differentiate implicitly with respect to $x$

   Apply the product rule to each side:

   $y'\log x+\frac{y}{x}=\log y+x\cdot\frac{y'}{y}$

   Step 3 — Collect the $y'$ terms

   $y'\log x-\frac{x\,y'}{y}=\log y-\frac{y}{x}$

   $y'\left(\log x-\frac{x}{y}\right)=\log y-\frac{y}{x}$

   Step 4 — Solve for $y'$ and tidy the fractions

   $y'=\frac{\log y-\dfrac{y}{x}}{\log x-\dfrac{x}{y}}=\frac{\dfrac{x\log y-y}{x}}{\dfrac{y\log x-x}{y}}=\frac{y\,(x\log y-y)}{x\,(y\log x-x)}$ …

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3. TG EAPCET 2025Set eng-2025-05-03-AN1 markMCQ
   Q.If the perpendicular distance from the focus of an ellipse $\dfrac{x^2}{9} + \dfrac{y^2}{b^2} = 1$ $(b < 3)$ to its corresponding directrix is $\dfrac{4}{\sqrt{5}}$, then the slope of the tangent to this ellipse drawn at $\left(\dfrac{3}{\sqrt{2}}, \dfrac{b}{\sqrt{2}}\right)$ is (A) $-\dfrac{2}{3}$ (B) $\dfrac{2}{3}$ (C) $\dfrac{3}{2}$ (D) $-\dfrac{3}{2}$

   ›Reveal solutionSolution

   $e=\dfrac{\sqrt5}{3},\;b^2=4$; the tangent slope at the given point is $-\dfrac{2}{3}$.

   For $\dfrac{x^2}{9}+\dfrac{y^2}{b^2}=1$ with $a=3$, the focus-to-directrix distance is

   $\frac{a}{e}-ae=\frac{a(1-e^2)}{e}=\frac{4}{\sqrt5}.$

   With $a=3$: $\;3\sqrt5\,e^2+4e-3\sqrt5=0\Rightarrow e=\dfrac{\sqrt5}{3}.$

   Then $e^2=\tfrac59$ and $b^2=a^2(1-e^2)=9\!\left(\tfrac49\right)=4$, so the ellipse is $\dfrac{x^2}{9}+\dfrac{y^2}{4}=1$. …

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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 $\cos5\theta$:

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

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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: …

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6. CA Foundation 2025Set sep-20251 markMCQ
   Q.Find $\dfrac{dy}{dx}$ for $x^2y^2 + y = 0$. (A) $\dfrac{dy}{dx} = \dfrac{2y^2x}{2y^2x^2 + 1}$ (B) $\dfrac{dy}{dx} = \dfrac{-2y^2x}{2yx^2 + 1}$ (C) $\dfrac{dy}{dx} = \dfrac{-2y^2x + 1}{2y^2x^2}$ (D) $\dfrac{dy}{dx} = \dfrac{2y^2x - 1}{2y^2x^2}$

   ›Reveal solutionSolution

   Implicit differentiation of $x^2y^2+y=0$ gives $\dfrac{dy}{dx} = \dfrac{-2xy^2}{2x^2y+1}$.

   Step 1 — Differentiate term by term (y depends on x)

   For $x^2y^2$ use the product rule:

   $\frac{d}{dx}(x^2y^2) = 2x\,y^2 + x^2\cdot 2y\frac{dy}{dx} = 2xy^2 + 2x^2y\frac{dy}{dx}$

   and $\dfrac{d}{dx}(y) = \dfrac{dy}{dx}$.

   Step 2 — Assemble the differentiated equation

   $2xy^2 + 2x^2y\frac{dy}{dx} + \frac{dy}{dx} = 0$

   Step 3 — Collect and solve for dy/dx

   $\frac{dy}{dx}\big(2x^2y + 1\big) = -2xy^2$

   $\frac{dy}{dx} = \frac{-2xy^2}{2x^2y + 1}$

   Why the other options are wrong: (A) drops the minus sign; (C) and (D) wrongly move the "+1" into the numerator, which happens only if you fail to factor $\frac{dy}{dx}$ correctly. …

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7. TG EAPCET 2024Set eng-2024-05-10-FN1 markMCQ
   Q.If $y = a \cos 3x + b e^{-x}$, then $y''(3 \sin 3x - \cos 3x) =$ (A) $10y' \sin 3x + 3y(\sin 3x + 3\cos 3x)$ (B) $10y' \cos 3x + 3y(\sin 3x + 3\cos 3x)$ (C) $10y' \cos 3x + 3y(\cos 3x + 3\sin 3x)$ (D) $10y' \cos 3x + 3y(\sin 3x - 3\cos 3x)$

   ›Reveal solutionSolution

   The key idea is to compute the first and second derivatives of $y = a \cos 3x + b e^{-x}$, then substitute into the expression $y''(3 \sin 3x - \cos 3x)$ and simplify to match one of the given forms. The result simplifies to $10y' \cos 3x + 3y(\sin 3x + 3\cos 3x)$, which corresponds to option (B).

   We start with the given function:

   $y = a \cos 3x + b e^{-x}$

   We need to find an expression for $y''(3 \sin 3x - \cos 3x)$ in terms of $y$ and $y'$. The trick is to avoid solving for $a$ and $b$ explicitly — instead, we differentiate and then cleverly combine terms.

   ---

   1. Compute the first derivative $y'$

   Differentiate term by term:

   $y' = -3a \sin 3x - b e^{-x}$

   (Recall: derivative of $\cos 3x$ is $-3\sin 3x$, and derivative of $e^{-x}$ is $-e^{-x}$.)

   2. Compute the second derivative $y''$

   Differentiate $y'$:

   $y'' = -9a \cos 3x + b e^{-x}$

   (Derivative of $-3a \sin 3x$ is $-9a \cos 3x$; derivative of $-b e^{-x}$ is $+b e^{-x}$ because $-e^{-x}$ differentiates to $e^{-x}$.)

   3. Form the expression $y''(3 \sin 3x - \cos 3x)$

   Substitute $y''$:

   $y''(3 \sin 3x - \cos 3x) = (-9a \cos 3x + b e^{-x})(3 \sin 3x - \cos 3x)$

   Expand:

   $= -27a \cos 3x \sin 3x + 9a \cos^2 3x + 3b e^{-x} \sin 3x - b e^{-x} \cos 3x$

   4. Express everything in terms of $y$ and $y'$

   We have:

   $y = a \cos 3x + b e^{-x}$

   $y' = -3a \sin 3x - b e^{-x}$

   We want to rewrite the expanded expression. Notice the terms:

   • $9a \cos^2 3x$ can be linked to $y$ and $y'$ if we also have $\sin^2$ terms, but we don't. Instead, we aim to match the pattern in the options, which involve $y' \cos 3x$ and $y(\sin 3x + 3\cos 3x)$.

   5. Try to express $a \cos 3x$ and $b e^{-x}$ from $y$ and $y'$

   From $y$ and $y'$ we can solve:

   $y = a \cos 3x + b e^{-x}$

   $y' = -3a \sin 3x - b e^{-x}$

   Add them:

   $y + y' = a \cos 3x - 3a \sin 3x = a(\cos 3x - 3\sin 3x)$

   So:

   $a = \frac{y + y'}{\cos 3x - 3\sin 3x}$

   Also subtract:

   $y - y' = a \cos 3x + 3a \sin 3x + 2b e^{-x}$

   But this gets messy. Instead, a better approach: directly compute the target expression using derivatives and compare to options.

   6. Compute $y''$ in terms of $y$ and $y'$

   Notice that from $y' = -3a \sin 3x - b e^{-x}$, we can differentiate again but also note:

   $y'' = -9a \cos 3x + b e^{-x}$

   We can write $y''$ as a combination of $y$ and $y'$:

   $y = a \cos 3x + b e^{-x}$

   $y' = -3a \sin 3x - b e^{-x}$

   Multiply $y$ by something and $y'$ by something to get $y''$. Try:

   $y'' = -9a \cos 3x + b e^{-x} = \alpha y + \beta y'$

   Substitute:

   $\alpha (a \cos 3x + b e^{-x}) + \beta (-3a \sin 3x - b e^{-x}) = a(\alpha \cos 3x - 3\beta \sin 3x) + b e^{-x}(\alpha - \beta)$

   Compare with $y'' = -9a \cos 3x + b e^{-x}$:

   • Coefficient of $a \cos 3x$: $\alpha = -9$
   • Coefficient of $a \sin 3x$: $-3\beta = 0 \Rightarrow \beta = 0$
   • Coefficient of $b e^{-x}$: $\alpha - \beta = -9 = 1$? That gives $-9 = 1$, impossible.

   So $y''$ is not a linear combination of $y$ and $y'$ alone (because the sine and cosine terms have different dependencies). So we must work directly.

   7. Instead, compute $y''(3\sin 3x - \cos 3x)$ by substituting $y''$ and then replace $a \cos 3x$ and $b e^{-x}$ using $y$ and $y'$

   We have:

   $y''(3\sin 3x - \cos 3x) = (-9a \cos 3x + b e^{-x})(3\sin 3x - \cos 3x)$

   Expand:

   $= -27a \cos 3x \sin 3x + 9a \cos^2 3x + 3b e^{-x} \sin 3x - b e^{-x} \cos 3x$

   Now, note that $y' = -3a \sin 3x - b e^{-x}$, so $3a \sin 3x = -y' - b e^{-x}$. But we have $-27a \cos 3x \sin 3x = -9 \cos 3x (3a \sin 3x) = -9 \cos 3x (-y' - b e^{-x}) = 9y' \cos 3x + 9b e^{-x} \cos 3x$.

   So the expression becomes:

   $= 9y' \cos 3x + 9b e^{-x} \cos 3x + 9a \cos^2 3x + 3b e^{-x} \sin 3x - b e^{-x} \cos 3x$

   Combine the $b e^{-x} \cos 3x$ terms: $9b e^{-x} \cos 3x - b e^{-x} \cos 3x = 8b e^{-x} \cos 3x$.

   So:

   $= 9y' \cos 3x + 9a \cos^2 3x + 3b e^{-x} \sin 3x + 8b e^{-x} \cos 3x$

   8. Now rewrite $9a \cos^2 3x$

   We know $y = a \cos 3x + b e^{-x}$, so $a \cos 3x = y - b e^{-x}$. Then:

   $9a \cos^2 3x = 9 \cos 3x (a \cos 3x) = 9 \cos 3x (y - b e^{-x}) = 9y \cos 3x - 9b e^{-x} \cos 3x$

   Substitute back:

   $= 9y' \cos 3x + (9y \cos 3x - 9b e^{-x} \cos 3x) + 3b e^{-x} \sin 3x + 8b e^{-x} \cos 3x$

   Combine the $b e^{-x} \cos 3x$: $-9 + 8 = -1$, so we get $-b e^{-x} \cos 3x$.

   Thus:

   $= 9y' \cos 3x + 9y \cos 3x + 3b e^{-x} \sin 3x - b e^{-x} \cos 3x$

   9. Factor $b e^{-x}$ terms

   Notice $3b e^{-x} \sin 3x - b e^{-x} \cos 3x = b e^{-x} (3 \sin 3x - \cos 3x)$. But $b e^{-x} = y - a \cos 3x$. However, we can also write $b e^{-x} = y - a \cos 3x$, but we want everything in $y$ and $y'$. Alternatively, note that $y' = -3a \sin 3x - b e^{-x}$ doesn't directly give $b e^{-x}$ alone.

   Instead, observe that $3y \sin 3x = 3a \cos 3x \sin 3x + 3b e^{-x} \sin 3x$. And $9y \cos 3x$ we already have. Let's try to match the options: they all have $10y' \cos 3x + 3y(\text{something})$.

   10. Compare with option (B): $10y' \cos 3x + 3y(\sin 3x + 3\cos 3x)$

   Expand option (B):

   $10y' \cos 3x + 3y \sin 3x + 9y \cos 3x$

   Our expression is:

   $9y' \cos 3x + 9y \cos 3x + 3b e^{-x} \sin 3x - b e^{-x} \cos 3x$

   We need to turn $9y' \cos 3x$ into $10y' \cos 3x$ and the remaining terms into $3y \sin 3x$. That suggests $y' \cos 3x$ appears with coefficient 10, so we might have missed a $y' \cos 3x$ term. Let's re-check step 7: we had $9y' \cos 3x$ from the manipulation, but perhaps there's an extra $y' \cos 3x$ hidden.

   11. Alternative direct substitution

   Let’s compute $y''(3\sin 3x - \cos 3x)$ by writing $y''$ in terms of $y$ and $y'$ using the original differential equation. Since $y = a \cos 3x + b e^{-x}$, note that $y$ satisfies a linear ODE. Differentiate twice: $y' = -3a \sin 3x - b e^{-x}$, $y'' = -9a \cos 3x + b e^{-x}$. Add $9y$:

   $y'' + 9y = (-9a \cos 3x + b e^{-x}) + 9(a \cos 3x + b e^{-x}) = 10 b e^{-x}$ …

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8. 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 $\theta = mxy$: …

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9. 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. …

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

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

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11. 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 $\sin\theta=\cos\theta=2^{-1/2}$: $\frac{dx}{d\theta}=-3\left(\frac12\right)\left(\frac{2}{\sqrt2}\right)=-3\cdot\frac12\cdot\sqrt2=-\frac{3\sqrt2}{2}.$ …

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12. 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 $\frac{dy}{dx}$

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

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

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