Q.Derivative of $x^2$ w.r.t. $x^3$ is __________.

Showing the 12 most recent of 48 on this concept.

1. AP EAPCET 2021Set eng-2021-08-23-FN1 markMCQ
   Q.If $y = \log\left(\dfrac{1+x}{1-x}\right)^{1/4} - \dfrac{1}{2}\tan^{-1}(x)$, then $\dfrac{dy}{dx}$ at $x = \dfrac{1}{\sqrt{2}}$ equals ______ (A) $\dfrac{-4}{3}$ (B) $\dfrac{4}{3}$ (C) $\dfrac{-2}{3}$ (D) $\dfrac{2}{3}$

   ›Reveal solutionSolution

   Tests differentiating a log-of-a-power expression combined with an arctan term, then evaluating at a specific point.

   Concept and Intuition

   Splitting $\log\left[\dfrac{1+x}{1-x}\right]^{1/4}$ using log rules turns it into $\dfrac14[\log(1+x)-\log(1-x)]$, which differentiates term-by-term far more easily than trying to apply the chain rule to the whole power-of-a-quotient directly.

   Step-by-Step Solution

   1. Rewrite $y=\dfrac14\log\left(\dfrac{1+x}{1-x}\right)-\dfrac12\tan^{-1}x = \dfrac14[\log(1+x)-\log(1-x)]-\dfrac12\tan^{-1}x$.
   2. Differentiate: $\dfrac{dy}{dx}=\dfrac14\left[\dfrac{1}{1+x}+\dfrac{1}{1-x}\right]-\dfrac12\cdot\dfrac{1}{1+x^2}$.
   3. Combine the bracket: $\dfrac{1}{1+x}+\dfrac{1}{1-x}=\dfrac{(1-x)+(1+x)}{1-x^2}=\dfrac{2}{1-x^2}$.
   4. So $\dfrac{dy}{dx}=\dfrac14\cdot\dfrac{2}{1-x^2}-\dfrac{1}{2(1+x^2)}=\dfrac{1}{2(1-x^2)}-\dfrac{1}{2(1+x^2)}$.
   5. Combine over a common denominator: $=\dfrac12\cdot\dfrac{(1+x^2)-(1-x^2)}{(1-x^2)(1+x^2)}=\dfrac12\cdot\dfrac{2x^2}{1-x^4}=\dfrac{x^2}{1-x^4}$. …

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2. AP EAPCET 2021Set eng-2021-08-24-AN1 markMCQ
   Q.The derivative of $\text{Sec}^{-1}\left(\dfrac{1}{2x^2-1}\right)$ with respect to $\sqrt{1-x^2}$ at $x=\dfrac{1}{2}$ equals ________ (A) 2 (B) $\dfrac{1}{2}$ (C) $\dfrac{1}{4}$ (D) 4

   ›Reveal solutionSolution

   Substituting $x=\cos\theta$ reduces both functions to simple multiples of $\theta$, giving the derivative of one with respect to the other as $2/x$, which equals $4$ at $x=1/2$.

   Concept and Intuition

   When asked for the derivative of one function of $x$ with respect to another function of $x$ (parametric-style differentiation), the trick is $\dfrac{du}{dv} = \dfrac{du/dx}{dv/dx}$. Here, substituting $x=\cos\theta$ turns the ugly $\operatorname{Sec}^{-1}\!\left(\dfrac{1}{2x^2-1}\right)$ into the clean double-angle expression $2\theta$.

   Step-by-Step Solution

   1. Let $x=\cos\theta$, $\theta\in[0,\pi]$, so $\sin\theta=\sqrt{1-x^2}\ge0$.
   2. Then $2x^2-1 = 2\cos^2\theta-1=\cos2\theta$, so $\dfrac{1}{2x^2-1}=\sec2\theta$.
   3. So $u=\operatorname{Sec}^{-1}(\sec2\theta)=2\theta = 2\operatorname{Cos}^{-1}x$ (valid since at $x=1/2$, $2\theta=2\pi/3\in[0,\pi]\setminus\{\pi/2\}$, the correct principal range for $\operatorname{Sec}^{-1}$).
   4. $\dfrac{du}{dx} = 2\cdot\left(\dfrac{-1}{\sqrt{1-x^2}}\right) = \dfrac{-2}{\sqrt{1-x^2}}$.
   5. $v=\sqrt{1-x^2}$, so $\dfrac{dv}{dx} = \dfrac{-x}{\sqrt{1-x^2}}$. …

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3. AP EAPCET 2024Set eng-2024-05-22-AN1 markMCQ
   Q.If $f(x)=5\cos^3x-3\sin^2x$ and $g(x)=4\sin^3x+\cos^2x$, then the derivative of $f(x)$ with respect to $g(x)$ is (A) $\dfrac{5\cos x+2}{6\cos x-1}$ (B) $-\left(\dfrac{5\cos x+2}{6\cos x-1}\right)$ (C) $\dfrac{15\cos x-6}{12\sin x+2}$ (D) $-\left(\dfrac{15\cos x+6}{12\sin x-2}\right)$

   ›Reveal solutionSolution

   The derivative of one function with respect to another is the ratio of their derivatives w.r.t. $x$; simplifying gives $-\dfrac{15\cos x+6}{12\sin x-2}$.

   Concept and Intuition

   When asked for $\dfrac{df}{dg}$ (derivative of $f$ with respect to $g$, not $x$), the chain rule gives $\dfrac{df}{dg}=\dfrac{df/dx}{dg/dx}$, valid wherever $g'(x)\ne0$. So computing both ordinary derivatives w.r.t. $x$ and dividing solves it directly.

   Step-by-Step Solution

   1. $f(x)=5\cos^3x-3\sin^2x$. $f'(x)=5\cdot3\cos^2x(-\sin x)-3\cdot2\sin x\cos x=-15\cos^2x\sin x-6\sin x\cos x=-3\sin x\cos x(5\cos x+2)$.
   2. $g(x)=4\sin^3x+\cos^2x$. $g'(x)=4\cdot3\sin^2x\cos x+2\cos x(-\sin x)=12\sin^2x\cos x-2\sin x\cos x=2\sin x\cos x(6\sin x-1)$. …

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4. AP EAPCET 2025Set eng-2025-05-24-FN1 markMCQ
   Q.If $y = \sqrt{\dfrac{x^4\sqrt{3x-5}}{(x^2-3)(2x-3)}}$, then $\left(\dfrac{dy}{dx}\right)_{x=2} =$ (A) $5$ (B) $0$ (C) $1$ (D) $-5$

   ›Reveal solutionSolution

   Logarithmic differentiation converts the messy root/product/quotient into a sum of logs; evaluating at $x=2$ gives $y'=-5$.

   Concept and Intuition

   Whenever a function is a product, quotient, or power of several simpler factors (especially under a square root), logarithmic differentiation is the cleanest route: take $\log$ of both sides to convert products to sums and powers to multiples, differentiate termwise, then multiply back by $y$.

   Step-by-Step Solution

   1. Write $y=\left[\dfrac{x^4\sqrt{3x-5}}{(x^2-3)(2x-3)}\right]^{1/2}$. Taking $\log$:

   $\log y = \frac12\Big[4\log x + \frac12\log(3x-5) - \log(x^2-3) - \log(2x-3)\Big].$

   2. Differentiate both sides with respect to $x$:

   $\frac{y'}{y}=\frac12\left[\frac{4}{x}+\frac12\cdot\frac{3}{3x-5}-\frac{2x}{x^2-3}-\frac{2}{2x-3}\right].$

   3. Evaluate the original $y$ at $x=2$: $x^4=16$, $3x-5=1\Rightarrow\sqrt{1}=1$, $x^2-3=1$, $2x-3=1$. So $y=\sqrt{16\cdot1/(1\cdot1)}=\sqrt{16}=4$.
   4. Evaluate each bracket term at $x=2$:
      • $\dfrac{4}{x}=\dfrac{4}{2}=2$
      • $\dfrac12\cdot\dfrac{3}{3x-5}=\dfrac12\cdot\dfrac{3}{1}=1.5$ …

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5. AP EAPCET 2025Set eng-2025-05-21-FN1 markMCQ
   Q.The derivative of $\mathrm{Sec}^{-1}\left(\dfrac{1}{2x^2-1}\right)$ with respect to $\sqrt{1-x^2}$ at $x=\dfrac{1}{2}$ is (A) $-2$ (B) $1$ (C) $2$ (D) $4$

   ›Reveal solutionSolution

   A "derivative with respect to another function" problem — the substitution $x=\cos\theta$ collapses $\mathrm{Sec}^{-1}(2x^2-1)^{-1}$ into $2\theta$, making the ratio of derivatives trivial. Answer: $4$.

   Concept and Intuition

   To find $\dfrac{du}{dv}$ where $u=u(x)$ and $v=v(x)$, use $\dfrac{du}{dv}=\dfrac{du/dx}{dv/dx}$ (or, more cleanly here, a common parameter $\theta$). The key trick recognizing $2x^2-1$ as $\cos2\theta$ when $x=\cos\theta$ turns the inverse secant of a rational expression into a simple linear function of $\theta$.

   Step-by-Step Solution

   1. Let $x=\cos\theta$, $\theta\in[0,\pi]$ (the natural domain for $\cos^{-1}$).
   2. Then $2x^2-1 = 2\cos^2\theta-1=\cos2\theta$, so $\dfrac{1}{2x^2-1}=\dfrac{1}{\cos2\theta}=\sec2\theta$.
   3. Hence $u=\mathrm{Sec}^{-1}(\sec 2\theta)$. Since $x=\tfrac12\Rightarrow\theta=\cos^{-1}\!\left(\tfrac12\right)=\tfrac{\pi}{3}$, we get $2\theta=\tfrac{2\pi}{3}$, which lies in $[0,\pi]$ — the principal range of $\mathrm{Sec}^{-1}$ — so $u=2\theta=2\cos^{-1}x$ validly (no branch correction needed here).
   4. Also $v=\sqrt{1-x^2}=\sqrt{1-\cos^2\theta}=\sin\theta$ (non-negative since $\theta\in[0,\pi]$).
   5. Differentiate w.r.t. $\theta$: $\dfrac{du}{d\theta}=2$, $\dfrac{dv}{d\theta}=\cos\theta=x$.
   6. So $\dfrac{du}{dv}=\dfrac{du/d\theta}{dv/d\theta}=\dfrac{2}{x}$.
   7. At $x=\dfrac12$: $\dfrac{du}{dv}=\dfrac{2}{1/2}=4$. …

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6. AP EAPCET 2021Set eng-2021-08-20-FN1 markMCQ
   Q.$\dfrac{d}{dx}\left\{\sin^2\left(\mathrm{Cot}^{-1}\sqrt{\dfrac{1+x}{1-x}}\right)\right\} =$ (A) 0 (B) $\dfrac{1}{2}$ (C) $\dfrac{-1}{2}$ (D) $-1$

   ›Reveal solutionSolution

   This tests simplifying an inverse-trig composite using the identity $\sin^2\theta=\tfrac{1}{1+\cot^2\theta}$ before differentiating, avoiding messy chain-rule work. Answer: $-\tfrac12$.

   Concept and Intuition

   Rather than differentiating $\sin^2(\mathrm{Cot}^{-1}(\cdots))$ directly through the chain rule, it's far simpler to algebraically simplify the whole expression to a function of $x$ first, since $\cot\theta$ is given explicitly.

   Step-by-Step Solution

   1. Let $\theta = \mathrm{Cot}^{-1}\sqrt{\dfrac{1+x}{1-x}}$, so $\cot\theta = \sqrt{\dfrac{1+x}{1-x}}$, hence $\cot^2\theta = \dfrac{1+x}{1-x}$.
   2. Using $\sin^2\theta = \dfrac{1}{1+\cot^2\theta}$: $\sin^2\theta = \dfrac{1}{1+\frac{1+x}{1-x}} = \dfrac{1-x}{(1-x)+(1+x)} = \dfrac{1-x}{2}$. …

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7. AP EAPCET 2025Set eng-2025-05-24-FN1 markMCQ
   Q.If $y = \mathrm{Tan}^{-1}\left(\dfrac{3x - x^3}{1 - 3x^2}\right) + \mathrm{Tan}^{-1}\left(\dfrac{7x}{1 - 12x^2}\right)$, then at $x = 0$, $\dfrac{dy}{dx} =$ (A) $6$ (B) $7$ (C) $9$ (D) $10$

   ›Reveal solutionSolution

   Differentiate the sum of two arctangent expressions term-by-term and evaluate at $x=0$; the answer is $10$.

   Concept and Intuition

   Rather than trying to recognize the whole expression as some multiple-angle identity, it is safest and fastest to differentiate each $\mathrm{Tan}^{-1}(\cdot)$ term directly using the chain rule $\dfrac{d}{dx}\mathrm{Tan}^{-1}(u)=\dfrac{u'}{1+u^2}$, then substitute $x=0$. Since we only need the derivative at a point, we don't need the general antiderivative simplification (e.g., recognizing $\frac{3x-x^3}{1-3x^2}$ as $\tan(3\theta)$ for $x=\tan\theta$) — direct differentiation is more robust.

   Step-by-Step Solution

   1. Let $y = \mathrm{Tan}^{-1}(u) + \mathrm{Tan}^{-1}(v)$ where $u=\dfrac{3x-x^3}{1-3x^2}$ and $v=\dfrac{7x}{1-12x^2}$.
   2. First term derivative at $x=0$:

   $u'=\frac{(3-3x^2)(1-3x^2)-(3x-x^3)(-6x)}{(1-3x^2)^2}.$

   At $x=0$: numerator $=(3)(1)-0=3$, denominator $=1$, so $u'(0)=3$. Also $u(0)=0$.

   Contribution: $\dfrac{u'(0)}{1+u(0)^2}=\dfrac{3}{1}=3$.

   3. Second term derivative at $x=0$:

   $v'=\frac{7(1-12x^2)-7x(-24x)}{(1-12x^2)^2}.$

   At $x=0$: numerator $=7(1)-0=7$, denominator $=1$, so $v'(0)=7$. Also $v(0)=0$. …

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8. AP EAPCET 2021Set eng-2021-08-24-FN1 markMCQ
   Q.The point on the curve $y = x^3$, at which the tangent to the curve is parallel to the x-axis, is ____ (A) $(2,2)$ (B) $(3,3)$ (C) $(4,4)$ (D) $(0,0)$

   ›Reveal solutionSolution

   A tangent parallel to the x-axis has zero slope; solving $y'=0$ for $y=x^3$ gives the point $(0,0)$.

   Concept and Intuition

   The slope of the tangent to a curve at a point equals the derivative there. "Parallel to the x-axis" means this slope is zero.

   Step-by-Step Solution

   1. $y=x^3\Rightarrow \dfrac{dy}{dx}=3x^2$.
   2. Set $3x^2=0\Rightarrow x=0$.
   3. At $x=0$, $y=0^3=0$, so the point is $(0,0)$. …

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9. AP EAPCET 2021Set eng-2021-08-19-FN1 markMCQ
   Q.If $y = x + \dfrac{1}{x}$, then which among the following holds? (A) $x^2 y' + xy = 0$ (B) $x^2 y' + xy + 2 = 0$ (C) $x^2 y' - xy + 2 = 0$ (D) $x^2 y' + xy - 2 = 0$

   ›Reveal solutionSolution

   Differentiating $y=x+1/x$ and combining $x^2y'$ with $xy$ eliminates $x$ entirely, leaving the identity $x^2y'-xy+2=0$.

   Concept and Intuition

   When a relation is asked "which of the following holds," the trick is to differentiate the given function and then algebraically combine $y$, $y'$, and $x$ to see which combination becomes a pure constant (eliminating $x$) — that combination is the required identity.

   Step-by-Step Solution

   1. $y=x+\dfrac1x \Rightarrow y'=1-\dfrac{1}{x^2}$.
   2. Multiply by $x^2$: $x^2y' = x^2-1$.
   3. Compute $xy = x\left(x+\dfrac1x\right)=x^2+1$.
   4. Subtract: $x^2y'-xy = (x^2-1)-(x^2+1)=-2$. …

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10. AP EAPCET 2021Set eng-2021-08-20-FN1 markMCQ
    Q.If $f(x) = 2x^2 + 3x - 5$, then the value of $f'(0) + 3f'(-1)$ is equal to _______ (A) 1 (B) 0 (C) 3 (D) 2

    ›Reveal solutionSolution

    This tests basic polynomial differentiation and evaluation at points. Answer: $0$.

    Concept and Intuition

    Differentiate the polynomial term by term using the power rule, then substitute the given values directly.

    Step-by-Step Solution

    1. $f(x)=2x^2+3x-5 \Rightarrow f'(x) = 4x+3$.
    2. $f'(0) = 4(0)+3 = 3$.
    3. $f'(-1) = 4(-1)+3 = -4+3=-1$. …

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11. AP EAPCET 2026Set eng-2026-05-15-AN1 markMCQ
    Q.If $f(x) = \mathrm{Tan}^{-1}\left(\frac{\sqrt{1+x^2} + \sqrt{1-x^2}}{\sqrt{1+x^2} - \sqrt{1-x^2}}\right)$, then $f'\left(-\frac{1}{2}\right) =$ (A) $-\frac{1}{\sqrt{2}}$ (B) $\frac{1}{\sqrt{2}}$ (C) $-\frac{2}{\sqrt{15}}$ (D) $\frac{2}{\sqrt{15}}$

    ›Reveal solutionSolution

    This tests simplifying an inverse-trig expression via a trigonometric substitution before differentiating. Once simplified, $f(x)=\frac\pi4+\frac12\cos^{-1}(x^2)$, giving $f'\left(-\frac12\right)=\frac2{\sqrt{15}}$.

    Concept and Intuition

    Expressions with $\sqrt{1+x^2}$ and $\sqrt{1-x^2}$ together strongly suggest substituting $x^2=\cos\varphi$, turning both square roots into half-angle sine/cosine forms via $1\pm\cos\varphi=2\cos^2\frac\varphi2$ or $2\sin^2\frac\varphi2$. This collapses the arctangent of a ratio into a simple tangent addition, making the function (and its derivative) far easier to handle than direct differentiation of the original expression.

    Step-by-Step Solution

    1. Let $x^2=\cos\varphi$. Then $1+x^2=1+\cos\varphi=2\cos^2\frac\varphi2$ and $1-x^2=1-\cos\varphi=2\sin^2\frac\varphi2$.
    2. So $\sqrt{1+x^2}=\sqrt2\cos\frac\varphi2$ and $\sqrt{1-x^2}=\sqrt2\sin\frac\varphi2$ (taking the principal positive roots).
    3. The ratio inside $f$: $\frac{\cos\frac\varphi2+\sin\frac\varphi2}{\cos\frac\varphi2-\sin\frac\varphi2}=\frac{1+\tan\frac\varphi2}{1-\tan\frac\varphi2}=\tan\left(\frac\pi4+\frac\varphi2\right)$.
    4. So $f(x)=\tan^{-1}\left[\tan\left(\frac\pi4+\frac\varphi2\right)\right]=\frac\pi4+\frac\varphi2=\frac\pi4+\frac12\cos^{-1}(x^2)$. …

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12. AP EAPCET 2023Set eng-2023-05-17-AN1 markMCQ
    Q.If the tangent drawn to the curve $y=x^3$ at a point $(\alpha,\beta)$ cuts again the curve at another point $(\alpha_1,\beta_1)$, then $\dfrac{\beta_1}{\beta}=$ (A) $-2$ (B) 1 (C) $-8$ (D) 27

    ›Reveal solutionSolution

    The tangent to $y=x^3$ at $x=\alpha$ meets the curve again at $x=-2\alpha$ — a classic result — so $\beta_1/\beta=(-2\alpha)^3/\alpha^3=-8$.

    Concept and Intuition

    Setting the tangent line equal to the curve gives a cubic in $x$ with a double root at the point of tangency (since the line touches there) and one more simple root at the second intersection. Factoring out $(x-\alpha)^2$ reveals that root directly.

    Step-by-Step Solution

    1. Tangent at $(\alpha,\alpha^3)$: slope $=3\alpha^2$, so $y=3\alpha^2(x-\alpha)+\alpha^3=3\alpha^2x-2\alpha^3$.
    2. Intersection with $y=x^3$: $x^3=3\alpha^2x-2\alpha^3\Rightarrow x^3-3\alpha^2x+2\alpha^3=0$.
    3. Since the tangent touches at $x=\alpha$, $(x-\alpha)^2$ divides this cubic: $x^3-3\alpha^2x+2\alpha^3=(x-\alpha)^2(x+2\alpha)$. …

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