Q.Differentiate $a^x$ w.r.t. $x$, where $a$ is a positive constant.

Showing the 12 most recent of 23 on this concept.

1. AP EAPCET 2026Set eng-2026-05-14-FN1 markMCQ
   Q.If $f(x) = \sqrt{2^{2x}\log(3x-2)}$, then $f'(2) =$ (A) $\dfrac{4\log 2 \log 4 + 3}{\sqrt{\log 4}}$ (B) $\dfrac{8(\log 2)^2 + 3}{2\sqrt{\log 4}}$ (C) $\dfrac{8(\log 4)^2 + 3}{2\sqrt{\log 4}}$ (D) $\dfrac{8\log 2 \log 4 + 3}{2\sqrt{\log 4}}$

   ›Reveal solutionSolution

   Differentiate $f(x)=\sqrt{2^{2x}\log(3x-2)}$ using the chain rule on the square root and the product rule inside; at $x=2$ this evaluates to $\dfrac{8\log2\log4+3}{2\sqrt{\log4}}$.

   Concept and Intuition

   Whenever a function is a square root of a product, write $f=\sqrt L$ so $f' = \dfrac{L'}{2\sqrt L}$ (chain rule), then find $L'$ using the product rule since $L(x)=2^{2x}\cdot\log(3x-2)$ is a product of an exponential and a log term. This two-layer differentiation is the key technique.

   Step-by-Step Solution

   1. Let $L(x) = 2^{2x}\log(3x-2)$, so $f(x)=\sqrt{L(x)}$ and $f'(x) = \dfrac{L'(x)}{2\sqrt{L(x)}}$.
   2. Evaluate $L(2)$: $2^{2(2)}=2^4=16$; $\log(3(2)-2)=\log 4$. So $L(2) = 16\log 4$, and $\sqrt{L(2)} = \sqrt{16\log4} = 4\sqrt{\log4}$.
   3. Find $L'(x)$ by the product rule: $L'(x) = \dfrac{d}{dx}\left[2^{2x}\right]\log(3x-2) + 2^{2x}\cdot\dfrac{d}{dx}\left[\log(3x-2)\right]$.
   4. $\dfrac{d}{dx}2^{2x} = 2^{2x}\cdot\ln2\cdot 2 = 2\cdot2^{2x}\log2$ (using $\log$ as natural log consistently), i.e. $2^{2x+1}\log2$.
   5. $\dfrac{d}{dx}\log(3x-2) = \dfrac{3}{3x-2}$.
   6. So $L'(x) = 2^{2x+1}\log2\cdot\log(3x-2) + 2^{2x}\cdot\dfrac{3}{3x-2}$.
   7. At $x=2$: $2^{2x+1} = 2^5=32$, $\log(3x-2)=\log4$, so first term $=32\log2\log4$. Second term: $2^{2x}=16$, $\dfrac{3}{3x-2}=\dfrac{3}{4}$, so second term $=16\times\dfrac34=12$.
   8. $L'(2) = 32\log2\log4+12$.
   9. $f'(2) = \dfrac{32\log2\log4+12}{2\times4\sqrt{\log4}} = \dfrac{32\log2\log4+12}{8\sqrt{\log4}} = \dfrac{8\log2\log4+3}{2\sqrt{\log4}}$ (dividing numerator and denominator by 4).

   Common Mistakes

   • Forgetting the extra factor of $2$ from differentiating $2^{2x}$ (chain rule on the exponent $2x$).
   • Not simplifying the final fraction by dividing by the common factor of 4, leading to an unmatched-looking but equivalent expression.

   ✓Final answer

   The correct option is (D) — $\dfrac{8\log 2 \log 4 + 3}{2\sqrt{\log 4}}$.

   ANSWER: D

2. AP EAPCET 2026Set eng-2026-05-14-FN1 markMCQ
   Q.If $y = \mathrm{Sec}^{-1}\left(\dfrac{1+x^2}{2x}\right)$ and $x > 1$, then $\dfrac{dy}{dx} =$ (A) $\dfrac{1}{1+x^2}$ (B) $\dfrac{2}{1+x^2}$ (C) $-\dfrac{1}{1+x^2}$ (D) $-\dfrac{2}{1+x^2}$

   ›Reveal solutionSolution

   Differentiating the inverse secant of a rational expression using the chain rule and careful algebraic simplification gives $\dfrac{dy}{dx}=\dfrac{2}{1+x^2}$.

   Concept and Intuition

   For $y=\mathrm{Sec}^{-1}(u)$, the derivative formula is $\dfrac{dy}{dx} = \dfrac{1}{|u|\sqrt{u^2-1}}\cdot\dfrac{du}{dx}$. Here $u=\dfrac{1+x^2}{2x}$ is positive for $x>1$, so we can drop the absolute value and just carefully simplify the algebra — the key insight is recognizing that $u^2-1$ factors as a perfect square-like expression involving $(x^2-1)^2$, which simplifies the square root beautifully.

   Step-by-Step Solution

   1. Let $u = \dfrac{1+x^2}{2x}$. Compute $\dfrac{du}{dx}$ using the quotient rule: $u' = \dfrac{2x(2x) - (1+x^2)(2)}{(2x)^2} = \dfrac{4x^2-2-2x^2}{4x^2} = \dfrac{2x^2-2}{4x^2} = \dfrac{x^2-1}{2x^2}$.
   2. Compute $u^2-1$: $u^2-1 = \dfrac{(1+x^2)^2-4x^2}{4x^2} = \dfrac{(1+x^2-2x)(1+x^2+2x)}{4x^2} = \dfrac{(x-1)^2(x+1)^2}{4x^2}$.
   3. So $\sqrt{u^2-1} = \dfrac{|x-1||x+1|}{2|x|} = \dfrac{|x^2-1|}{2|x|}$. For $x>1$: $x^2-1>0$ and $x>0$, so $\sqrt{u^2-1}=\dfrac{x^2-1}{2x}$.
   4. For $x>1$, $u = \dfrac{1+x^2}{2x}>0$, so $|u|=u=\dfrac{1+x^2}{2x}$.
   5. $u\sqrt{u^2-1} = \dfrac{1+x^2}{2x}\cdot\dfrac{x^2-1}{2x} = \dfrac{(1+x^2)(x^2-1)}{4x^2}$.
   6. $\dfrac{dy}{dx} = \dfrac{u'}{u\sqrt{u^2-1}} = \dfrac{(x^2-1)/(2x^2)}{(1+x^2)(x^2-1)/(4x^2)} = \dfrac{x^2-1}{2x^2}\times\dfrac{4x^2}{(1+x^2)(x^2-1)} = \dfrac{4}{2(1+x^2)} = \dfrac{2}{1+x^2}$.

   Common Mistakes

   • Dropping the absolute value carelessly without checking the sign of $x$ in the given domain $x>1$.
   • Forgetting to simplify $u^2-1$ using the difference-of-squares-of-squares factoring trick, leading to a messy, harder-to-simplify square root.

   ✓Final answer

   The correct option is (B) — $\dfrac{2}{1+x^2}$.

   ANSWER: B

3. AP EAPCET 2026Set eng-2026-05-15-FN1 markMCQ
   Q.If $x = \mathrm{Sinh}^{-1}t + \log(t^2 + 1)$ and $y = \mathrm{Tan}^{-1}t + \log|t|$, then $\dfrac{dy}{dx} =$ (A) $\dfrac{t^2 + t + 1}{2t + \sqrt{t^4 + t^2}}$ (B) $\dfrac{t^2 + t + 1}{2t + \sqrt{t^2 + 1}}$ (C) $\dfrac{t^2 + t + 1}{2t^2 + \sqrt{t^4 + t^2}}$ (D) $\dfrac{t^2 + t + 1}{2 + \sqrt{1 + t^2}}$

   ›Reveal solutionSolution

   This tests parametric differentiation $\left(\frac{dy}{dx}=\frac{dy/dt}{dx/dt}\right)$ with inverse-hyperbolic and inverse-trig terms; the answer is option (C).

   Concept and Intuition

   When both $x$ and $y$ are given in terms of a parameter $t$, we find $dy/dx$ as the ratio of the two derivatives with respect to $t$, using $\frac{d}{dt}\mathrm{Sinh}^{-1}t=\frac{1}{\sqrt{1+t^2}}$ and $\frac{d}{dt}\mathrm{Tan}^{-1}t=\frac{1}{1+t^2}$.

   Step-by-Step Solution

   1. $\dfrac{dx}{dt} = \dfrac{1}{\sqrt{1+t^2}} + \dfrac{2t}{t^2+1}$ (since $\frac{d}{dt}\log(t^2+1)=\frac{2t}{t^2+1}$).
   2. $\dfrac{dy}{dt} = \dfrac{1}{1+t^2} + \dfrac1t = \dfrac{t + (1+t^2)}{t(1+t^2)} = \dfrac{t^2+t+1}{t(1+t^2)}$.
   3. Write $\dfrac{dx}{dt}$ over common denominator $1+t^2$: $\dfrac{dx}{dt} = \dfrac{\sqrt{1+t^2}+2t}{1+t^2}$.
   4. $\dfrac{dy}{dx} = \dfrac{dy/dt}{dx/dt} = \dfrac{t^2+t+1}{t(1+t^2)}\cdot\dfrac{1+t^2}{\sqrt{1+t^2}+2t} = \dfrac{t^2+t+1}{t\left(\sqrt{1+t^2}+2t\right)} = \dfrac{t^2+t+1}{t\sqrt{1+t^2}+2t^2}$.
   5. Since $t\sqrt{1+t^2}=\sqrt{t^2(1+t^2)}=\sqrt{t^4+t^2}$, this is $\dfrac{t^2+t+1}{2t^2+\sqrt{t^4+t^2}}$.

   Common Mistakes

   • Forgetting the chain-rule factor $2t/(t^2+1)$ in $\log(t^2+1)$.
   • Leaving the denominator in the un-simplified form $t\sqrt{1+t^2}+2t^2$ instead of recognizing it matches $\sqrt{t^4+t^2}$.

   ✓Final answer

   The correct option is (C) — $\dfrac{t^2 + t + 1}{2t^2 + \sqrt{t^4 + t^2}}$.

   ANSWER: C

4. AP EAPCET 2026Set eng-2026-05-18-AN1 markMCQ
   Q.If $y = \operatorname{Sech}^{-1}\left(\dfrac{9}{9x^2 + 10}\right)$, then $\dfrac{dy}{dx} =$ (A) $\dfrac{-18x}{\sqrt{(9x^2+10)^2 + 81}}$ (B) $\dfrac{-18x}{\sqrt{(9x^2+10)^2 - 81}}$ (C) $\dfrac{18x}{\sqrt{(9x^2+19)(9x^2+1)}}$ (D) $\dfrac{18x(9x^2+10)}{\sqrt{(9x^2+19)(9x^2+1)}}$

   ›Reveal solutionSolution

   A chain-rule differentiation of an inverse hyperbolic function, where $(9x^2+10)^2-81$ factors as a difference of squares into $(9x^2+1)(9x^2+19)$.

   Concept and Intuition

   For $y=\operatorname{sech}^{-1}u$, the standard derivative is $\dfrac{dy}{du}=\dfrac{-1}{u\sqrt{1-u^2}}$ (for $0<u<1$). The chain rule then just needs $du/dx$, and the algebra simplifies neatly because $(9x^2+10)^2-9^2$ is a difference of squares.

   Step-by-Step Solution

   1. Let $u=\dfrac{9}{9x^2+10}$. Then $\dfrac{du}{dx}=9\cdot\dfrac{-18x}{(9x^2+10)^2}=\dfrac{-162x}{(9x^2+10)^2}$.
   2. $1-u^2=1-\dfrac{81}{(9x^2+10)^2}=\dfrac{(9x^2+10)^2-81}{(9x^2+10)^2}$.
   3. Factor as a difference of squares: $(9x^2+10)^2-9^2=(9x^2+10-9)(9x^2+10+9)=(9x^2+1)(9x^2+19)$.
   4. So $\sqrt{1-u^2}=\dfrac{\sqrt{(9x^2+1)(9x^2+19)}}{9x^2+10}$.
   5. $u\sqrt{1-u^2}=\dfrac{9}{9x^2+10}\cdot\dfrac{\sqrt{(9x^2+1)(9x^2+19)}}{9x^2+10}=\dfrac{9\sqrt{(9x^2+1)(9x^2+19)}}{(9x^2+10)^2}$.
   6. $\dfrac{dy}{dx}=\dfrac{-1}{u\sqrt{1-u^2}}\cdot\dfrac{du}{dx}=\dfrac{-(9x^2+10)^2}{9\sqrt{(9x^2+1)(9x^2+19)}}\cdot\dfrac{-162x}{(9x^2+10)^2}=\dfrac{162x}{9\sqrt{(9x^2+1)(9x^2+19)}}=\dfrac{18x}{\sqrt{(9x^2+1)(9x^2+19)}}$.

   Common Mistakes

   • Missing the sign cancellation between the $-1$ in the $\operatorname{sech}^{-1}$ derivative formula and the negative $du/dx$, which would flip the final sign.
   • Not recognizing $(9x^2+10)^2-81$ as a factorable difference of squares and instead leaving it unsimplified (masking the match with the given options).

   ✓Final answer

   The correct option is (C) — $\dfrac{18x}{\sqrt{(9x^2+19)(9x^2+1)}}$.

   ANSWER: C

5. AP EAPCET 2025Set eng-2025-05-21-AN1 markMCQ
   Q.If $y = \tanh^{-1}\sqrt{\dfrac{1-x}{1+x}}$, then $\dfrac{dy}{dx} =$ (A) $-\dfrac{1}{2\sqrt{1-x^2}}$ (B) $-\dfrac{1}{2x\sqrt{1-x^2}}$ (C) $\dfrac{2}{1+x^2}$ (D) $\dfrac{1}{2x\sqrt{1+x^2}}$

   ›Reveal solutionSolution

   Differentiating $\tanh^{-1}u$ via the chain rule and simplifying $u(1+x)=\sqrt{1-x^2}$ gives $dy/dx=-\dfrac{1}{2x\sqrt{1-x^2}}$.

   Concept and Intuition

   $\tanh^{-1}z=\dfrac12\log\dfrac{1+z}{1-z}$ has derivative $\dfrac{1}{1-z^2}$, exactly like a standard log-based inverse function. Here $z=u(x)$ is itself a composite square-root expression, so the chain rule applies twice; the algebra simplifies nicely because $u^2$ is a simple rational function of $x$.

   Step-by-Step Solution

   1. Let $u=\sqrt{\dfrac{1-x}{1+x}}$, so $y=\tanh^{-1}u$ and $\dfrac{dy}{du}=\dfrac{1}{1-u^2}$.
   2. $u^2=\dfrac{1-x}{1+x}\Rightarrow 1-u^2=1-\dfrac{1-x}{1+x}=\dfrac{(1+x)-(1-x)}{1+x}=\dfrac{2x}{1+x}$.
   3. Differentiate $u^2=\dfrac{1-x}{1+x}$ w.r.t. $x$: $2u\dfrac{du}{dx}=\dfrac{-(1+x)-(1-x)}{(1+x)^2}=\dfrac{-2}{(1+x)^2}\Rightarrow \dfrac{du}{dx}=\dfrac{-1}{u(1+x)^2}$.
   4. By the chain rule: $\dfrac{dy}{dx}=\dfrac{dy}{du}\cdot\dfrac{du}{dx}=\dfrac{1+x}{2x}\cdot\left(\dfrac{-1}{u(1+x)^2}\right)=\dfrac{-1}{2x\,u(1+x)}$.
   5. Simplify $u(1+x)$: $u(1+x)=\sqrt{\dfrac{1-x}{1+x}}\cdot(1+x)=\sqrt{(1-x)(1+x)}=\sqrt{1-x^2}$.
   6. So $\dfrac{dy}{dx}=\dfrac{-1}{2x\sqrt{1-x^2}}$.

   Common Mistakes

   • Forgetting the extra factor of $x$ that survives from $\dfrac{1}{1-u^2}=\dfrac{1+x}{2x}$, leading to option (A)'s answer (missing the $x$ in the denominator).
   • Sign errors differentiating $\dfrac{1-x}{1+x}$ via the quotient rule.

   ✓Final answer

   The correct option is (B) — $-\dfrac{1}{2x\sqrt{1-x^2}}$.

   ANSWER: B

6. AP EAPCET 2025Set eng-2025-05-22-FN1 markMCQ
   Q.If $x = 2\cos^3\theta$ and $y = 3\sin^2\theta$, then $\frac{dy}{dx} =$ (A) $-\sec\theta$ (B) $\cos\theta$ (C) $-\mathrm{cosec}\,\theta$ (D) $\sin\theta$

   ›Reveal solutionSolution

   Differentiate $x$ and $y$ separately with respect to the parameter $\theta$, then take the ratio. Answer: $-\sec\theta$.

   Concept and Intuition

   For a parametric curve $x=x(\theta),\ y=y(\theta)$, the derivative is found via the chain rule as $\dfrac{dy}{dx}=\dfrac{dy/d\theta}{dx/d\theta}$, avoiding the need to eliminate $\theta$ and differentiate implicitly.

   Step-by-Step Solution

   1. $x=2\cos^3\theta$. Differentiate: $\dfrac{dx}{d\theta}=2\cdot3\cos^2\theta\cdot(-\sin\theta)=-6\cos^2\theta\sin\theta$.
   2. $y=3\sin^2\theta$. Differentiate: $\dfrac{dy}{d\theta}=3\cdot2\sin\theta\cos\theta=6\sin\theta\cos\theta$.
   3. $\dfrac{dy}{dx}=\dfrac{dy/d\theta}{dx/d\theta}=\dfrac{6\sin\theta\cos\theta}{-6\cos^2\theta\sin\theta}$.
   4. Cancel the common factor $6\sin\theta\cos\theta$ (assuming $\sin\theta,\cos\theta\ne0$): $=\dfrac{1}{-\cos\theta}=-\sec\theta$.

   Common Mistakes

   • Forgetting the chain-rule factor when differentiating $\cos^3\theta$ or $\sin^2\theta$ (missing the "3" or "2" power-rule multiplier).
   • Sign error when cancelling $\sin\theta\cos\theta$ from numerator and denominator, dropping the leading minus sign.

   ✓Final answer

   The correct option is (A) — $-\sec\theta$.

   ANSWER: A

7. AP EAPCET 2025Set eng-2025-05-23-AN1 markMCQ
   Q.If $y = \sqrt{\cosh x + \sqrt{\cosh x}}$, then $\dfrac{dy}{dx} =$ (A) $\dfrac{\sinh x (2y^2 + 2\cosh x + 1)}{4y(y^2 + \cosh x)}$ (B) $\dfrac{\sinh x (2y^2 - 2\cosh x - 1)}{4y(y^2 - \cosh x)}$ (C) $\dfrac{\sinh x (1 - 2\sqrt{\cosh x})}{4y\sqrt{\cosh x}}$ (D) $\dfrac{\sinh x (1 + 2\sqrt{\cosh x})}{4y\sqrt{\cosh x}}$

   ›Reveal solutionSolution

   Square both sides to remove the outer root, differentiate implicitly, then

   tidy the resulting fraction — the answer comes out directly in terms of $y$

   and $\cosh x$, matching option (D).

   Concept and Intuition

   When $y$ is defined as a nested square root, it's usually easier to square first

   ($y^2=$ the inside) and differentiate implicitly rather than applying the chain

   rule twice directly to the nested radical — this avoids stacking two

   $\tfrac{1}{2\sqrt{\cdot}}$ factors and keeps the algebra manageable.

   Step-by-Step Solution

   1. $y=\sqrt{\cosh x+\sqrt{\cosh x}}\ \Rightarrow\ y^2=\cosh x+\sqrt{\cosh x}$.
   2. Differentiate both sides w.r.t. $x$: $2y\,\dfrac{dy}{dx}=\sinh x+\dfrac{1}{2\sqrt{\cosh x}}\cdot\sinh x=\sinh x\left(1+\dfrac{1}{2\sqrt{\cosh x}}\right)$.
   3. Combine the bracket over a common denominator: $1+\dfrac{1}{2\sqrt{\cosh x}}=\dfrac{2\sqrt{\cosh x}+1}{2\sqrt{\cosh x}}$.
   4. So $2y\,y'=\dfrac{\sinh x\,(2\sqrt{\cosh x}+1)}{2\sqrt{\cosh x}}$.
   5. Solve for $y'$: $y'=\dfrac{\sinh x\,(1+2\sqrt{\cosh x})}{4y\sqrt{\cosh x}}$.

   Common Mistakes

   • Differentiating the nested radical directly without squaring first, which tangles two chain-rule layers and often drops a factor of 2.
   • Forgetting to carry the $2y$ from the implicit differentiation (i.e., leaving the answer in terms of $y^2$ instead of solving for $y'$ explicitly).

   ✓Final answer

   The correct option is (D) — $\dfrac{\sinh x(1+2\sqrt{\cosh x})}{4y\sqrt{\cosh x}}$.

   ANSWER: D

8. AP EAPCET 2025Set eng-2025-05-23-AN1 markMCQ
   Q.If $y = \mathrm{Tan}^{-1}\sqrt{x^2 - 1} + \mathrm{Sinh}^{-1}\sqrt{x^2 - 1}$, $x > 1$, then $\dfrac{dy}{dx} =$ (A) $\dfrac{1}{x\sqrt{x^2-1}}$ (B) $\dfrac{x+1}{x\sqrt{x^2-1}}$ (C) $\dfrac{x+1}{x^2\sqrt{x^2-1}}$ (D) $\dfrac{x}{\sqrt{x^2-1}}$

   ›Reveal solutionSolution

   Both inverse-function derivatives share the same inner derivative

   $u'=x/\sqrt{x^2-1}$; the $1+u^2$ under $\mathrm{Tan}^{-1}$ and $\sqrt{1+u^2}$ under

   $\mathrm{Sinh}^{-1}$ both simplify beautifully because $u^2=x^2-1$, so $1+u^2=x^2$.

   Concept and Intuition

   Both $\mathrm{Tan}^{-1}$ and $\mathrm{Sinh}^{-1}$ have derivative formulas built around

   $1+u^2$ (as $\dfrac{1}{1+u^2}$ and $\dfrac{1}{\sqrt{1+u^2}}$ respectively). Here

   $u=\sqrt{x^2-1}$ makes $1+u^2=x^2$ exactly, a clean perfect square — this is why

   the two inverse functions are paired together in the problem, since they

   combine so tidily.

   Step-by-Step Solution

   1. Let $u=\sqrt{x^2-1}$. Then $u'=\dfrac{x}{\sqrt{x^2-1}}$ and $1+u^2 = 1+(x^2-1)=x^2$.
   2. $\dfrac{d}{dx}\mathrm{Tan}^{-1}u=\dfrac{u'}{1+u^2}=\dfrac{x/\sqrt{x^2-1}}{x^2}=\dfrac{1}{x\sqrt{x^2-1}}$.
   3. $\dfrac{d}{dx}\mathrm{Sinh}^{-1}u=\dfrac{u'}{\sqrt{1+u^2}}=\dfrac{x/\sqrt{x^2-1}}{\sqrt{x^2}}=\dfrac{x/\sqrt{x^2-1}}{x}=\dfrac{1}{\sqrt{x^2-1}}$ (since $x>1$ means $\sqrt{x^2}=x$, not $|x|$ ambiguity).
   4. Add: $\dfrac{dy}{dx}=\dfrac{1}{x\sqrt{x^2-1}}+\dfrac{1}{\sqrt{x^2-1}}=\dfrac{1+x}{x\sqrt{x^2-1}}$.

   Common Mistakes

   • Using $\sqrt{x^2}=|x|$ and leaving an unnecessary absolute value, when the domain $x>1$ already fixes the sign.
   • Misremembering the derivative of $\mathrm{Sinh}^{-1}u$ as $\dfrac{1}{1+u^2}$ (that's $\mathrm{Tan}^{-1}$'s formula) instead of $\dfrac{1}{\sqrt{1+u^2}}$.

   ✓Final answer

   The correct option is (B) — $\dfrac{x+1}{x\sqrt{x^2-1}}$.

   ANSWER: B

9. AP EAPCET 2025Set eng-2025-05-26-AN1 markMCQ
   Q.If $f(x) = \sec^{-1}\left(\dfrac{1}{2x^2-1}\right)$ and $g(x) = \tan^{-1}\left(\dfrac{\sqrt{1+x^2}-1}{x}\right)$, then the derivative of $f(x)$ with respect to $g(x)$ is (A) $\dfrac{1+x^2}{4\sqrt{1-x^2}}$ (B) $\dfrac{1-x^2}{4\sqrt{1+x^2}}$ (C) $-\dfrac{4(1-x^2)}{\sqrt{1+x^2}}$ (D) $-\dfrac{4(1+x^2)}{\sqrt{1-x^2}}$

   ›Reveal solutionSolution

   Simplify both $f$ and $g$ using standard inverse-trig substitutions, then take the ratio of derivatives. Answer: $\dfrac{df}{dg}=-\dfrac{4(1+x^2)}{\sqrt{1-x^2}}$.

   Concept and Intuition

   $\dfrac{df}{dg}$ means $\dfrac{df/dx}{dg/dx}$. Both $f$ and $g$ are compositions of inverse trig functions that simplify beautifully with the right substitution ($x=\cos\theta$-type for $f$, $x=\tan\theta$ for $g$), turning ugly inverse-trig expressions into simple linear multiples of $\tan^{-1}x$ or $\cos^{-1}x$.

   Step-by-Step Solution

   1. $f(x)=\sec^{-1}\left(\dfrac{1}{2x^2-1}\right)=\cos^{-1}(2x^2-1)$ since $\sec^{-1}(1/u)=\cos^{-1}u$.
   2. For $x\in(0,1)$, write $x=\cos\theta$: then $2x^2-1=2\cos^2\theta-1=\cos2\theta$, so $f=\cos^{-1}(\cos2\theta)=2\theta=2\cos^{-1}x$.
   3. $f'(x) = 2\cdot\left(-\dfrac{1}{\sqrt{1-x^2}}\right) = -\dfrac{2}{\sqrt{1-x^2}}$.
   4. For $g$: put $x=\tan\theta$, so $\sqrt{1+x^2}=\sec\theta$. Then $\dfrac{\sqrt{1+x^2}-1}{x}=\dfrac{\sec\theta-1}{\tan\theta}=\dfrac{1-\cos\theta}{\sin\theta}=\tan\left(\dfrac{\theta}{2}\right)$.
   5. So $g(x)=\tan^{-1}\left(\tan\dfrac{\theta}{2}\right)=\dfrac{\theta}{2}=\dfrac12\tan^{-1}x$.
   6. $g'(x)=\dfrac{1}{2(1+x^2)}$.
   7. $\dfrac{df}{dg}=\dfrac{f'(x)}{g'(x)}=\dfrac{-2/\sqrt{1-x^2}}{1/[2(1+x^2)]} = -\dfrac{2}{\sqrt{1-x^2}}\cdot 2(1+x^2) = -\dfrac{4(1+x^2)}{\sqrt{1-x^2}}$.

   Common Mistakes

   • Missing that $\sec^{-1}(1/u)=\cos^{-1}u$ and instead trying to differentiate $\sec^{-1}$ of the messy argument directly.
   • Sign/half-angle slip when simplifying $\dfrac{1-\cos\theta}{\sin\theta}$ to $\tan(\theta/2)$.

   ✓Final answer

   The correct option is (D) — $-\dfrac{4(1+x^2)}{\sqrt{1-x^2}}$.

   ANSWER: D

10. AP EAPCET 2025Set eng-2025-05-27-FN1 markMCQ
    Q.If $x = \sqrt{2^{\operatorname{cosec}^{-1} t}}$ and $y = \sqrt{2^{\sec^{-1} t}}$, $|t| \geq 1$ then $\dfrac{dy}{dx} =$ (A) $\dfrac{x}{y}$ (B) $\dfrac{y}{x}$ (C) $-\dfrac{y}{x}$ (D) $-\dfrac{x}{y}$

    ›Reveal solutionSolution

    The identity $\operatorname{cosec}^{-1}t+\sec^{-1}t=\pi/2$ lets both $x$ and $y$ be written as exponentials of a single parameter $u$, so $dy/dx$ follows from parametric differentiation. Answer: $-\dfrac yx$.

    Concept and Intuition

    When $x$ and $y$ are both given as functions of a common (possibly hidden) parameter — here through the complementary inverse trig identity — the cleanest path is to introduce that parameter explicitly and use $\dfrac{dy}{dx}=\dfrac{dy/du}{dx/du}$, rather than trying to eliminate $t$ directly.

    Step-by-Step Solution

    1. For $|t|\ge1$, the standard identity $\operatorname{cosec}^{-1}t+\sec^{-1}t=\dfrac\pi2$ holds.
    2. Let $u=\operatorname{cosec}^{-1}t$. Then $\sec^{-1}t=\dfrac\pi2-u$.
    3. $x=\sqrt{2^u}=2^{u/2}$, and $y=\sqrt{2^{\pi/2-u}}=2^{(\pi/2-u)/2}=2^{\pi/4-u/2}$.
    4. Differentiate w.r.t. $u$: $\dfrac{dx}{du}=2^{u/2}\ln2\cdot\dfrac12=\dfrac{x\ln2}2$.
    5. $\dfrac{dy}{du}=2^{\pi/4-u/2}\ln2\cdot\left(-\dfrac12\right)=-\dfrac{y\ln2}2$.
    6. $\dfrac{dy}{dx}=\dfrac{dy/du}{dx/du}=\dfrac{-y\ln2/2}{x\ln2/2}=-\dfrac yx$.

    Common Mistakes

    • Forgetting the identity linking $\operatorname{cosec}^{-1}t$ and $\sec^{-1}t$ and instead trying to differentiate $x$ and $y$ directly with respect to $t$ using the (more complicated) derivative of $\operatorname{cosec}^{-1}t$.
    • Sign error: missing the minus sign that comes from $\sec^{-1}t=\pi/2-u$ (a decreasing function of $u$).

    ✓Final answer

    The correct option is (C) — $-\dfrac{y}{x}$.

    ANSWER: C

11. AP EAPCET 2024Set eng-2024-05-18-FN1 markMCQ
    Q.If $x=3\left[\sin t-\log\left(\cot\dfrac{t}{2}\right)\right]$ and $y=6\left[\cos t+\log\left(\tan\dfrac{t}{2}\right)\right]$ then $\dfrac{dy}{dx}=$ (A) $\dfrac{2\sin^2 t}{1+\sin t \cos t}$ (B) $\dfrac{2\cos^2 t}{1+\sin 2t}$ (C) $\dfrac{2\cos^2 t}{1+\sin t\cos t}$ (D) $\dfrac{1+\cos 2t}{1+\sin 2t}$

    ›Reveal solutionSolution

    A parametric-differentiation problem built around the classic identity $\dfrac{d}{dt}\log\tan(t/2)=\csc t$; the final simplified slope is $\dfrac{2\cos^2t}{1+\sin t\cos t}$.

    Concept and Intuition

    When $x$ and $y$ are given as functions of a parameter $t$, $\dfrac{dy}{dx}=\dfrac{dy/dt}{dx/dt}$. The log-tangent-half-angle term is a recurring building block whose derivative simplifies neatly to $\csc t$, which is worth memorizing to avoid a messy chain-rule expansion each time.

    Step-by-Step Solution

    1. Recall $\dfrac{d}{dt}\log\tan(t/2)=\dfrac{\sec^2(t/2)}{2\tan(t/2)}=\dfrac{1}{2\sin(t/2)\cos(t/2)}=\dfrac1{\sin t}=\csc t$. Hence $\dfrac{d}{dt}\log\cot(t/2)=-\csc t$.
    2. $x=3[\sin t-\log\cot(t/2)]\Rightarrow \dfrac{dx}{dt}=3[\cos t-(-\csc t)]=3(\cos t+\csc t)$.
    3. $y=6[\cos t+\log\tan(t/2)]\Rightarrow \dfrac{dy}{dt}=6[-\sin t+\csc t]$.
    4. $\dfrac{dy}{dx}=\dfrac{6(\csc t-\sin t)}{3(\cos t+\csc t)}=\dfrac{2(\csc t-\sin t)}{\cos t+\csc t}$.
    5. Simplify: $\csc t-\sin t=\dfrac{1-\sin^2t}{\sin t}=\dfrac{\cos^2t}{\sin t}$; and $\cos t+\csc t=\dfrac{\sin t\cos t+1}{\sin t}$.
    6. So $\dfrac{dy}{dx}=\dfrac{2\cos^2t/\sin t}{(1+\sin t\cos t)/\sin t}=\dfrac{2\cos^2t}{1+\sin t\cos t}$.

    Common Mistakes

    • Misremembering the sign in $\dfrac{d}{dt}\log\cot(t/2)$ (it is $-\csc t$, not $+\csc t$).
    • Not cancelling the common $\sin t$ factor from numerator and denominator, leaving an unsimplified answer that doesn't match any option.

    ✓Final answer

    The correct option is (C) — $\dfrac{2\cos^2 t}{1+\sin t\cos t}$.

    ANSWER: C

12. AP EAPCET 2024Set eng-2024-05-22-FN1 markMCQ
    Q.If $f(0) = 0$, $f'(0) = 3$, then the derivative of $y = f(f(f(f(f(x)))))$ at $x = 0$ is (A) $16$ (B) $32$ (C) $81$ (D) $243$

    ›Reveal solutionSolution

    Because $0$ is a fixed point of $f$ ($f(0)=0$), differentiating a repeated composition at $x=0$ just multiplies $f'(0)$ by itself once per composition — five times here.

    Concept and Intuition

    By the chain rule, $\dfrac{d}{dx}f(g(x))=f'(g(x))g'(x)$. For a chain of five compositions, the derivative at a point is a product of five factors of $f'$, each evaluated at the running value of the inner composition at that point. Since $f(0)=0$, the running value stays $0$ throughout, so every factor is just $f'(0)$.

    Step-by-Step Solution

    1. Let $y=f(f(f(f(f(x)))))$ (five nested $f$'s).
    2. By repeated chain rule: $y'(x)=f'\big(f(f(f(f(x))))\big)\cdot f'\big(f(f(f(x)))\big)\cdot f'\big(f(f(x))\big)\cdot f'\big(f(x)\big)\cdot f'(x)$.
    3. At $x=0$: since $f(0)=0$, we get $f(x)=0$, then $f(f(x))=f(0)=0$, and so on — every nested value at $x=0$ is $0$.
    4. So every one of the five factors becomes $f'(0)=3$.
    5. $y'(0)=3\times3\times3\times3\times3=3^5=243$.

    Common Mistakes

    • Forgetting that $f(0)=0$ is what makes every inner argument collapse to $0$; without this the chain rule would need each intermediate value separately.
    • Miscounting the number of compositions (four vs five nested $f$'s).

    ✓Final answer

    The correct option is (D) — $243$.

    ANSWER: D