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

Showing the 12 most recent of 13 on this concept.

1. COMEDK 2026Set 2026-A1 markMCQ
   Q.

   [!FORMULA] $\text { If } y=\tan ^{-1}\left(\frac{\sqrt{1+x^3}+\sqrt{1-x^3}}{\sqrt{1+x^3}-\sqrt{1-x^3}}\right) \text { then } \frac{\boldsymbol{d} \boldsymbol{y}}{\boldsymbol{d x}}=$

   (A) $-\frac{3 x^2}{2 \sqrt{1-x^6}}$ (B) $-\frac{6 x^2}{\sqrt{1-x^6}}$ (C) $\frac{6 x^2}{\sqrt{1-x^6}}$ (D) $\frac{3 x^2}{\sqrt{1-x^6}}$

   ›Reveal solutionSolution

   The key is to simplify the argument of the inverse tangent using the identity $\tan^{-1}\left(\frac{a+b}{a-b}\right) = \frac{\pi}{4} + \tan^{-1}\left(\frac{b}{a}\right)$, then differentiate. The derivative simplifies to $-\frac{3x^2}{2\sqrt{1-x^6}}$, so the correct option is (A).

   We start with

   $y = \tan^{-1}\left(\frac{\sqrt{1+x^3}+\sqrt{1-x^3}}{\sqrt{1+x^3}-\sqrt{1-x^3}}\right).$

   The expression inside looks messy, but there’s a classic trick: when you see a fraction of the form $\frac{A+B}{A-B}$, it often simplifies via the identity

   $\tan^{-1}\left(\frac{A+B}{A-B}\right) = \frac{\pi}{4} + \tan^{-1}\left(\frac{B}{A}\right),$

   provided $A > B > 0$ (which holds here for small $x$). This works because $\tan(\frac{\pi}{4}+\theta) = \frac{1+\tan\theta}{1-\tan\theta}$, and setting $\tan\theta = B/A$ gives exactly our fraction. This reduces the problem to differentiating a much simpler expression.

   1. Apply the identity Let $A = \sqrt{1+x^3}$ and $B = \sqrt{1-x^3}$. Then

   $y = \tan^{-1}\left(\frac{A+B}{A-B}\right) = \frac{\pi}{4} + \tan^{-1}\left(\frac{B}{A}\right).$

   Since $\frac{\pi}{4}$ is constant,

   $\frac{dy}{dx} = \frac{d}{dx} \tan^{-1}\left(\frac{B}{A}\right).$

   2. Simplify the ratio

   $\frac{B}{A} = \frac{\sqrt{1-x^3}}{\sqrt{1+x^3}} = \sqrt{\frac{1-x^3}{1+x^3}}.$

   So

   $y = \frac{\pi}{4} + \tan^{-1}\left( \sqrt{\frac{1-x^3}{1+x^3}} \right).$

   3. Differentiate using the chain rule Let $u = \sqrt{\frac{1-x^3}{1+x^3}}$. Then

   $\frac{dy}{dx} = \frac{1}{1+u^2} \cdot \frac{du}{dx}.$

   First compute $1+u^2$:

   $u^2 = \frac{1-x^3}{1+x^3}, \quad \text{so} \quad 1+u^2 = 1 + \frac{1-x^3}{1+x^3} = \frac{(1+x^3)+(1-x^3)}{1+x^3} = \frac{2}{1+x^3}.$

   Hence

   $\frac{1}{1+u^2} = \frac{1+x^3}{2}.$

   4. Find $\frac{du}{dx}$ Write $u = \left( \frac{1-x^3}{1+x^3} \right)^{1/2}$. Differentiate using the chain rule and quotient rule:

   $\frac{du}{dx} = \frac{1}{2} \left( \frac{1-x^3}{1+x^3} \right)^{-1/2} \cdot \frac{d}{dx}\left( \frac{1-x^3}{1+x^3} \right).$

   The derivative of the quotient:

   $\frac{d}{dx}\left( \frac{1-x^3}{1+x^3} \right) = \frac{(-3x^2)(1+x^3) - (1-x^3)(3x^2)}{(1+x^3)^2} = \frac{-3x^2(1+x^3 + 1 - x^3)}{(1+x^3)^2} = \frac{-6x^2}{(1+x^3)^2}.$

   Also note that

   $\left( \frac{1-x^3}{1+x^3} \right)^{-1/2} = \sqrt{\frac{1+x^3}{1-x^3}} = \frac{1}{u}.$

   So

   $\frac{du}{dx} = \frac{1}{2} \cdot \frac{1}{u} \cdot \frac{-6x^2}{(1+x^3)^2} = \frac{-3x^2}{u (1+x^3)^2}.$

   5. Combine to get $\frac{dy}{dx}$

   $\frac{dy}{dx} = \frac{1+x^3}{2} \cdot \frac{-3x^2}{u (1+x^3)^2} = \frac{-3x^2}{2u (1+x^3)}.$

   But $u = \sqrt{\frac{1-x^3}{1+x^3}}$, so $u(1+x^3) = \sqrt{1-x^3}\sqrt{1+x^3} = \sqrt{(1-x^3)(1+x^3)} = \sqrt{1-x^6}$.

   Therefore

   $\frac{dy}{dx} = -\frac{3x^2}{2\sqrt{1-x^6}}.$

   Watch out

   A common mistake is to forget the factor $\frac{1}{2}$ from the square root derivative, or to mishandle the algebra of $1+u^2$. Always simplify $1+u^2$ before plugging in $du/dx$ — it cancels nicely.

   Tip

   The identity $\tan^{-1}\left(\frac{A+B}{A-B}\right) = \frac{\pi}{4} + \tan^{-1}(B/A)$ is a powerful shortcut for symmetric fractions. It turns a complicated-looking derivative into a routine chain rule problem.

   ✓Final answer

   The correct option is (A).

   ANSWER: A

2. COMEDK 2025Set 2025-A1 markMCQ
   Q.Differentiate $\log _a x$ with respect to $a^x$ (A) $\frac{1}{x a^x}$ (B) $\frac{1}{x a^x(\log a)^2}$ (C) $\frac{a^x}{x(\log a)^2}$ (D) $\frac{a^x}{x}$

   ›Reveal solutionSolution

   We want the derivative of $\log_a x$ with respect to $a^x$. Using the chain rule in reverse (differentiating one function of a variable with respect to another function of the same variable), we get $\frac{1}{x a^x (\log a)^2}$, which corresponds to option (B).

   ---

   Concept & Intuition

   The phrase “differentiate $f$ with respect to $g$” means: treat $g$ as the independent variable and find $\frac{df}{dg}$. If both $f$ and $g$ are functions of a common variable (here $x$), we use the chain rule:

   $\frac{df}{dg} = \frac{df/dx}{dg/dx}.$

   So we compute the ordinary derivatives of $\log_a x$ and $a^x$ with respect to $x$, then take their ratio.

   ---

   Step-by-step solution

   1. Rewrite $\log_a x$ in terms of natural logs

   $\log_a x = \frac{\ln x}{\ln a}.$

   This is the standard change-of-base formula. Here $\ln a$ is a constant.

   2. Differentiate $\log_a x$ with respect to $x$

   $\frac{d}{dx} \left( \frac{\ln x}{\ln a} \right) = \frac{1}{\ln a} \cdot \frac{1}{x} = \frac{1}{x \ln a}.$

   3. Differentiate $a^x$ with respect to $x$

      Recall that $\frac{d}{dx} a^x = a^x \ln a$. (This comes from writing $a^x = e^{x \ln a}$ and using the chain rule.)

   4. Apply the “derivative with respect to” formula

   $\frac{d(\log_a x)}{d(a^x)} = \frac{ \frac{d}{dx}(\log_a x) }{ \frac{d}{dx}(a^x) } = \frac{ \frac{1}{x \ln a} }{ a^x \ln a }.$

   5. Simplify

   $\frac{1}{x \ln a} \cdot \frac{1}{a^x \ln a} = \frac{1}{x a^x (\ln a)^2}.$

   Since $\ln a$ and $\log a$ are the same (natural log), we can write $(\log a)^2$ in the denominator.

   Tip

   A common mistake is to forget the extra $\ln a$ in the denominator from differentiating $a^x$. Another is to confuse $\frac{d}{dx} \log_a x$ with $\frac{1}{x}$ — that’s only true for natural log. Always use $\frac{1}{x \ln a}$.

   ---

   ✓Final answer

   The correct option is (B).

   ANSWER: B

3. COMEDK 2025Set 2025-M1 markMCQ
   Q.If $f(x)=\left(\frac{3+x}{1+x}\right)^{2+3 x}$, then $f^{\prime}(0)=$ (A) $12+\log 3$ (B) $-12+3 \log 3$ (C) $-\frac{4}{3}+3 \log 3$ (D) $-12+27 \log 3$

   ›Reveal solutionSolution

   To differentiate a function of the form $h(x)^{g(x)}$, we use logarithmic differentiation: take $\log$, differentiate implicitly, then evaluate at $x=0$. The result is $f'(0) = -12 + 27 \log 3$, which corresponds to option (D).

   We have $f(x) = \left( \frac{3+x}{1+x} \right)^{2+3x}$. This is a variable base raised to a variable exponent — a classic case for logarithmic differentiation. The reason: neither the power rule nor the exponential rule alone applies, but taking logs converts the exponent into a factor, letting us use the product rule.

   ---

   1. Take the natural logarithm of both sides Let $y = f(x)$. Then

   $\log y = (2+3x) \log\left( \frac{3+x}{1+x} \right).$

   2. Differentiate implicitly with respect to $x$ On the left: $\frac{d}{dx} \log y = \frac{y'}{y}$. On the right: use the product rule. Let $u = 2+3x$ and $v = \log\left( \frac{3+x}{1+x} \right)$. Then

   $\frac{y'}{y} = u' v + u v'.$

   Here $u' = 3$.

   For $v$, note $\log\left( \frac{3+x}{1+x} \right) = \log(3+x) - \log(1+x)$, so

   $v' = \frac{1}{3+x} - \frac{1}{1+x}.$

   3. Write the derivative expression

   $\frac{y'}{y} = 3 \log\left( \frac{3+x}{1+x} \right) + (2+3x)\left( \frac{1}{3+x} - \frac{1}{1+x} \right).$

   Hence

   $y' = y \left[ 3 \log\left( \frac{3+x}{1+x} \right) + (2+3x)\left( \frac{1}{3+x} - \frac{1}{1+x} \right) \right].$

   4. Evaluate at $x=0$

      First, $y(0) = \left( \frac{3}{1} \right)^{2} = 9$.

      Next, $\log\left( \frac{3+0}{1+0} \right) = \log 3$.

      Also, $\frac{1}{3+0} - \frac{1}{1+0} = \frac{1}{3} - 1 = -\frac{2}{3}$.

      And $2+3(0) = 2$.

      So

   $f'(0) = 9 \left[ 3 \log 3 + 2 \left( -\frac{2}{3} \right) \right] = 9 \left[ 3 \log 3 - \frac{4}{3} \right].$

   5. Simplify

   $f'(0) = 27 \log 3 - 12.$

   Watch out

   A common mistake is forgetting to multiply by $y(0)=9$ after substituting into the bracket. Another is mis-simplifying $\frac{1}{3} - 1$ as $-\frac{2}{3}$ correctly but then losing the factor of 2 from $(2+3x)$.

   Tip

   Notice the final expression is $-12 + 27 \log 3$, which matches option (D). The order of terms doesn’t matter — it’s the same as $27 \log 3 - 12$.

   ✓Final answer

   The correct option is (D).

   ANSWER: D

4. COMEDK 2025Set 2025-M1 markMCQ
   Q.If $y=\sin ^{-1}\left(\frac{1}{\sqrt{x+1}}\right)$ then $\frac{d y}{d x}=$ (A) $\frac{1}{2 \sqrt{1-x}}$ (B) $\frac{1}{2 \sqrt{x}(1+\sqrt{x})}$ (C) $\frac{1}{2 x(1+\sqrt{x})}$ (D) $-\frac{1}{2 \sqrt{x}(1+x)}$

   ›Reveal solutionSolution

   The derivative simplifies by substituting $x = \sec^2\theta - 1$ or using a chain rule with algebraic manipulation; the final result is $-\frac{1}{2\sqrt{x}(1+x)}$, which matches option (D).

   We start with

   $y = \sin^{-1}\left(\frac{1}{\sqrt{x+1}}\right).$

   The argument $\frac{1}{\sqrt{x+1}}$ is always between 0 and 1 for $x \ge 0$, so the inverse sine is well-defined. The key insight: instead of differentiating directly and getting tangled in messy algebra, we can simplify the expression before differentiating by using a trigonometric substitution or by rewriting the argument in a friendlier form.

   ---

   1. Rewrite the argument Let $t = \frac{1}{\sqrt{x+1}}$. Then $t^2 = \frac{1}{x+1}$, so $x+1 = \frac{1}{t^2}$ and $x = \frac{1}{t^2} - 1$. But more directly: notice that

   $\frac{1}{\sqrt{x+1}} = \sqrt{\frac{1}{x+1}}.$

   This suggests setting $x+1 = \sec^2\theta$ (since $\sec\theta \ge 1$ for $\theta \in [0,\pi/2)$). Then

   $\frac{1}{\sqrt{x+1}} = \frac{1}{\sec\theta} = \cos\theta.$

   So

   $y = \sin^{-1}(\cos\theta).$

   2. Simplify the inverse trig expression Recall the identity: $\sin^{-1}(\cos\theta) = \frac{\pi}{2} - \theta$ for $\theta \in [0,\pi]$. Since $\theta = \sec^{-1}(\sqrt{x+1})$ and $\sqrt{x+1} \ge 1$, $\theta$ lies in $[0, \pi/2)$, so the identity holds. Hence

   $y = \frac{\pi}{2} - \theta = \frac{\pi}{2} - \sec^{-1}(\sqrt{x+1}).$

   3. Differentiate The derivative of $\sec^{-1}(u)$ is $\frac{1}{|u|\sqrt{u^2-1}} \cdot \frac{du}{dx}$. Here $u = \sqrt{x+1} > 0$, so the absolute value is unnecessary.

   $\frac{dy}{dx} = 0 - \frac{1}{\sqrt{x+1} \cdot \sqrt{(\sqrt{x+1})^2 - 1}} \cdot \frac{d}{dx}(\sqrt{x+1}).$

   Now $\frac{d}{dx}(\sqrt{x+1}) = \frac{1}{2\sqrt{x+1}}$.

   So

   $\frac{dy}{dx} = -\frac{1}{\sqrt{x+1} \cdot \sqrt{x+1 - 1}} \cdot \frac{1}{2\sqrt{x+1}}.$

   Simplify: $\sqrt{x+1 - 1} = \sqrt{x}$.

   Thus

   $\frac{dy}{dx} = -\frac{1}{\sqrt{x+1} \cdot \sqrt{x}} \cdot \frac{1}{2\sqrt{x+1}} = -\frac{1}{2\sqrt{x}\,(x+1)}.$

   4. Match with options The result is $-\frac{1}{2\sqrt{x}(1+x)}$, which is exactly option (D).

   Watch out

   A common mistake is forgetting the negative sign from the identity $\sin^{-1}(\cos\theta) = \frac{\pi}{2} - \theta$, or incorrectly simplifying $\sqrt{(\sqrt{x+1})^2 - 1}$ to $\sqrt{x}$ without checking domain.

   Tip

   The substitution $\frac{1}{\sqrt{x+1}} = \cos\theta$ turns the problem into a trivial derivative of $\frac{\pi}{2} - \theta$, avoiding messy chain-rule algebra.

   ✓Final answer

   The correct option is (D).

   ANSWER: D

5. KCET 2024Set A-11 markMCQ
   Q.Let the function satisfy the equation $f(x+y) = f(x)f(y)$ for all $x, y \in \mathbb{R}$, where $f(0) \neq 0$. If $f(5) = 3$ and $f'(0) = 2$, then $f'(5)$ is (A) $6$ (B) $0$ (C) $3$ (D) $-6$

   ›Reveal solutionSolution

   The functional equation $f(x+y)=f(x)f(y)$ with $f(0)\neq0$ forces $f$ to be an exponential function. Using the given $f(5)=3$ and $f'(0)=2$, we find $f'(5)=6$, so the answer is (A).

   The core idea here is that the equation $f(x+y)=f(x)f(y)$ is the Cauchy exponential functional equation. For functions continuous at even a single point (or differentiable at a point, as we have here), the only non-zero solutions are of the form $f(x)=a^x$ for some positive base $a$. But we don't need to assume continuity — differentiability at $0$ is enough to pin down the derivative everywhere.

   Let’s see why. The equation tells us that the value of $f$ at a sum is the product of its values at the parts. This is the defining property of exponential functions. If we differentiate with respect to $y$ and then set $y=0$, we get a direct relation between $f'(x)$ and $f(x)$.

   1. Differentiate the functional equation with respect to $y$. Treat $x$ as fixed. The left side is $f(x+y)$, whose derivative with respect to $y$ is $f'(x+y)$ (by the chain rule). The right side is $f(x)f(y)$, whose derivative is $f(x)f'(y)$. So:

   $f'(x+y) = f(x)f'(y)$

   This holds for all real $x$ and $y$.

   2. Set $y=0$. Then $f'(x+0) = f(x)f'(0)$. But $f'(0)=2$ is given, so:

   $f'(x) = 2\,f(x)$

   This is a beautiful result: the derivative of $f$ at any point is just twice the function’s value at that point. It tells us $f$ satisfies the differential equation $f' = 2f$, which indeed gives $f(x)=Ce^{2x}$.

   3. Find $f(0)$ using the functional equation.

      Put $x=y=0$: $f(0+0)=f(0)f(0)$, so $f(0)=[f(0)]^2$. Since $f(0)\neq0$, we can divide both sides by $f(0)$ to get $f(0)=1$. This is a standard check — any non-zero solution to this equation must have $f(0)=1$.

   4. Now compute $f'(5)$.

      From step 2, $f'(5)=2\,f(5)$. And we are given $f(5)=3$. So:

   $f'(5)=2 \times 3 = 6$

   Watch out

   A common mistake is to try to find $f(x)$ explicitly first by solving $f'=2f$ and then using $f(5)=3$ to get $f(x)=3e^{2(x-5)}$, then differentiate. That works, but it’s longer. The relation $f'(x)=2f(x)$ came directly from the functional equation and the given $f'(0)$ — no need to solve for $f$ at all.

   Tip

   The step $f'(x+y)=f(x)f'(y)$ followed by setting $y=0$ is a powerful trick for any functional equation where you have differentiability. It turns the functional property into a differential one instantly.

   ✓Final answer

   The value of $f'(5)$ is $\boxed{6}$, which corresponds to option (A).

6. COMEDK 2024Set 2024-A1 markMCQ
   Q.

   [!FORMULA] $\text { If } y=\sin ^{-1}(\sqrt{\sin x}) \text {, then } \frac{d y}{d x} \text { equals }$

   (A) $\frac{1}{2} \sqrt{1-\operatorname{cosec} x}$ (B) $\frac{1}{2} \sqrt{1-\sin x}$ (C) $\frac{1}{2} \sqrt{1+\operatorname{cosec} x}$ (D) $\frac{1}{2} \sqrt{1+\sin x}$

   ›Reveal solutionSolution

   Chain rule on $y=\sin^{-1}(\sqrt{\sin x})$ simplifies to $\frac{dy}{dx}=\frac12\sqrt{1+\operatorname{cosec}x}$ — option (C).

   Set up the chain rule

   Let $u=\sqrt{\sin x}$, so $y=\sin^{-1}u$ and $u^2=\sin x$.

   $\frac{dy}{dx}=\frac{1}{\sqrt{1-u^2}}\cdot\frac{du}{dx}.$

   Differentiate $u=(\sin x)^{1/2}$:

   $\frac{du}{dx}=\frac{\cos x}{2\sqrt{\sin x}}.$

   Since $u^2=\sin x$, we have $\sqrt{1-u^2}=\sqrt{1-\sin x}$, so

   $\frac{dy}{dx}=\frac{1}{\sqrt{1-\sin x}}\cdot\frac{\cos x}{2\sqrt{\sin x}}=\frac{\cos x}{2\sqrt{\sin x}\,\sqrt{1-\sin x}}.$

   Simplify with $\cos^2 x=(1-\sin x)(1+\sin x)$

   On the principal domain ($\cos x\ge0$), $\cos x=\sqrt{(1-\sin x)(1+\sin x)}$, hence

   $\frac{dy}{dx}=\frac{\sqrt{(1-\sin x)(1+\sin x)}}{2\sqrt{\sin x}\,\sqrt{1-\sin x}}=\frac{\sqrt{1+\sin x}}{2\sqrt{\sin x}}=\frac12\sqrt{\frac{1+\sin x}{\sin x}}=\frac12\sqrt{1+\operatorname{cosec}x}.$

   Numerical check at $x=\frac{\pi}{6}$: the formula gives $\frac12\sqrt{1+2}=\frac{\sqrt3}{2}\approx0.866$, matching the direct derivative.

   ✓Final answer

   $\dfrac{dy}{dx}=\dfrac12\sqrt{1+\operatorname{cosec}x}$. Correct option: (C).

   ANSWER: C

7. COMEDK 2024Set 2024-E1 markMCQ
   Q.

   [!FORMULA] $\text { If } y=\sin ^{-1}\left(\frac{5 x+12 \sqrt{1-x^2}}{13}\right) \text { then } \frac{d y}{d x} \text { equals }$

   (A) $\frac{-2 x}{\sqrt{1-x^2}}$ (B) $\frac{-1}{\sqrt{1+x^2}}$ (C) $\frac{1}{\sqrt{1-x^2}}$ (D) $\frac{2 x}{\sqrt{1-x^2}}$

   ›Reveal solutionSolution

   Substituting $x=\sin\theta$ turns the argument into $\sin(\theta+\phi)$, so $y=\sin^{-1}x+\phi$ and $\frac{dy}{dx}=\frac{1}{\sqrt{1-x^2}}$. The correct option is (C).

   Concept

   When an inverse-sine argument has the form $\frac{a x+b\sqrt{1-x^2}}{c}$ with $a^2+b^2=c^2$, the substitution $x=\sin\theta$ collapses $a\sin\theta+b\cos\theta$ into $c\sin(\theta+\phi)$, so the inverse sine unwinds to a simple sum of an angle and a constant — making differentiation immediate.

   Solution

   1. Substitute. $x=\sin\theta$, $\theta\in[-\tfrac{\pi}{2},\tfrac{\pi}{2}]$, so $\sqrt{1-x^2}=\cos\theta$ and

   $y=\sin^{-1}\!\left(\frac{5\sin\theta+12\cos\theta}{13}\right).$

   2. Single sine. With $\cos\phi=\tfrac{5}{13},\ \sin\phi=\tfrac{12}{13}$,

   $5\sin\theta+12\cos\theta=13\sin(\theta+\phi),\qquad y=\sin^{-1}\big(\sin(\theta+\phi)\big).$

   3. Unwind. On the principal branch $y=\theta+\phi=\sin^{-1}x+\phi$.
   4. Differentiate. Since $\phi$ is constant, $\dfrac{dy}{dx}=\dfrac{d}{dx}\sin^{-1}x=\dfrac{1}{\sqrt{1-x^2}}$.

   Tip

   Choosing $x=\sin\theta$ (so the extra term is $+\cos\theta$) keeps $y=\sin^{-1}x+\text{const}$, giving the $+\frac{1}{\sqrt{1-x^2}}$ branch that matches the options.

   ✓Final answer

   The correct option is (C) — $\dfrac{dy}{dx}=\dfrac{1}{\sqrt{1-x^2}}$.

8. COMEDK 2023Set 2023-E1 markMCQ
   Q.

   [!FORMULA] $\text { If } f(x)=\sin ^{-1}\left(\frac{2^{x+1}}{1+4^x}\right) \text { then } f^{\prime}(0) \text { is equal to }$

   (A) $2 \log 2$ (B) $\frac{2}{3} \log 2$ (C) 0 (D) $\log 2$

   ›Reveal solutionSolution

   Substituting $t=2^x$ turns the argument into $\frac{2t}{1+t^2}=\sin(2\tan^{-1}t)$, so $f(x)=2\tan^{-1}(2^x)$ and $f'(0)=\log 2$.

   Rewrite the argument with $t=2^x$:

   $\frac{2^{x+1}}{1+4^x}=\frac{2\cdot 2^x}{1+(2^x)^2}=\frac{2t}{1+t^2}.$

   Since $\frac{2t}{1+t^2}=\sin\big(2\tan^{-1}t\big)$, we get

   $f(x)=\sin^{-1}\!\Big(\frac{2t}{1+t^2}\Big)=2\tan^{-1}(2^x).$

   Differentiate:

   $f'(x)=2\cdot\frac{1}{1+(2^x)^2}\cdot 2^x\log 2=\frac{2\cdot 2^x\log 2}{1+4^x}.$

   At $x=0$ ($2^0=1$):

   $f'(0)=\frac{2(1)\log 2}{1+1}=\log 2=\log 2.$

   ✓Final answer

   The correct option is (D) — $\log 2$

9. KCET 2021Set A-11 markMCQ
   Q.If $y = (\cos x^2)^2$, then $\frac{dy}{dx}$ is equal to (A) $-4x \sin 2x^2$ (B) $-x \sin x^2$ (C) $-2x \sin 2x^2$ (D) $-x \cos 2x^2$

   ›Reveal solutionSolution

   Use the chain rule twice: differentiate the outer square, then the cosine, then the inner $x^2$. The derivative is $-4x \sin(x^2) \cos(x^2)$, which simplifies to $-2x \sin(2x^2)$.

   The function $y = (\cos x^2)^2$ is a composition of three layers: an outer square, a middle cosine, and an innermost $x^2$. Whenever you see a function of a function of a function, the chain rule is your tool — you differentiate from the outside in, multiplying each derivative along the way.

   A common mistake is to forget that $\cos x^2$ means $\cos(x^2)$, not $(\cos x)^2$. Here the parentheses make it clear: $(\cos x^2)^2$ means "square the cosine of $x^2$". So the outermost operation is squaring, then cosine, then $x^2$.

   Let’s work through it step by step.

   1. Identify the layers.

      Write $y = u^2$ where $u = \cos v$ and $v = x^2$.

      Then $\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dv} \cdot \frac{dv}{dx}$.

   2. Differentiate each layer.

      • $\frac{dy}{du} = 2u = 2\cos v = 2\cos(x^2)$
      • $\frac{du}{dv} = -\sin v = -\sin(x^2)$
      • $\frac{dv}{dx} = 2x$

   3. Multiply them together.

   $\frac{dy}{dx} = 2\cos(x^2) \cdot \bigl(-\sin(x^2)\bigr) \cdot 2x$

   $= -4x \,\cos(x^2)\,\sin(x^2)$

   4. Simplify using a trig identity. Recall the double-angle identity: $\sin 2\theta = 2\sin\theta\cos\theta$. Here $\theta = x^2$, so $2\sin(x^2)\cos(x^2) = \sin(2x^2)$. Therefore,

   $-4x \,\cos(x^2)\,\sin(x^2) = -2x \cdot \bigl(2\sin(x^2)\cos(x^2)\bigr) = -2x \sin(2x^2)$

   Tip

   Spotting the double-angle identity early saves a step. Once you have $-4x \sin(x^2)\cos(x^2)$, you can immediately write $-2x \sin(2x^2)$ — no need to expand further.

   Watch out

   A frequent error is to differentiate $\cos x^2$ as $-\sin x^2$ and stop, forgetting the chain rule on $x^2$ itself. That would give $-2x \sin x^2$, which is option (B) — a tempting distractor. Always check: did you multiply by the derivative of every inner function?

   Comparing with the options:

   • (A) $-4x \sin 2x^2$ — close, but the coefficient is double what it should be.
   • (B) $-x \sin x^2$ — missing the factor from the outer square and the inner derivative.
   • (C) $-2x \sin 2x^2$ — matches our result exactly.
   • (D) $-x \cos 2x^2$ — wrong function entirely.

   ✓Final answer

   The correct option is (C).

10. COMEDK 2021Set 2021-B1 markMCQ
    Q.If f(1)=3 and f'(1) = 4, then the value of the derivative of $\tan^{-1}[f(x)]$ at x = 1 is (A) 3/17 (B) 4/7 (C) 2/5 (D) 1/2

    ›Reveal solutionSolution

    The derivative at $x=1$ is $\tfrac25$.

    By the chain rule, $\dfrac{d}{dx}\tan^{-1}[f(x)]=\dfrac{f'(x)}{1+[f(x)]^2}$. Substituting $f(1)=3$, $f'(1)=4$:

    $\frac{4}{1+3^2}=\frac{4}{10}=\frac{2}{5}.$

    ✓Final answer

    The correct option is (C) — $\frac{2}{5}$

11. COMEDK 2021Set 2021-B1 markMCQ
    Q.If $f(x)=\log_{x^2}(\log_e x)$, then f'(x) at x = e is (A) infinite (B) 1/2e (C) 0 (D) 1/e

    ›Reveal solutionSolution

    $f'(e)=\dfrac{1}{2e}$.

    Convert to natural logs: $f(x)=\log_{x^2}(\log_e x)=\dfrac{\ln(\ln x)}{\ln(x^2)}=\dfrac{\ln(\ln x)}{2\ln x}$.

    Let $u=\ln(\ln x)$ and $v=2\ln x$, so $f=u/v$ and $f'=\dfrac{u'v-uv'}{v^2}$.

    $u'=\dfrac{1}{\ln x}\cdot\dfrac{1}{x}=\dfrac{1}{x\ln x}$, $\quad v'=\dfrac{2}{x}$.

    At $x=e$: $\ln e=1$, so $u=\ln(1)=0$, $v=2$, $u'=\dfrac{1}{e}$, $v'=\dfrac{2}{e}$.

    $f'(e)=\dfrac{(1/e)(2)-(0)(2/e)}{2^2}=\dfrac{2/e}{4}=\dfrac{1}{2e}.$

    ✓Final answer

    The correct option is (B) — $\dfrac{1}{2e}$

12. KCET 2019Set A-11 markMCQ
    Q.If $3\sqrt{y}\sqrt{x} = \sqrt[6]{(x+y)^5}$, then $\frac{dy}{dx} =$ (A) $\frac{x}{y}$ (B) $x+y$ (C) $x-y$ (D) $\frac{y}{x}$

    ›Reveal solutionSolution

    $\dfrac{dy}{dx} = \dfrac{y}{x}$ — option (D).

    The relation $\sqrt[3]{y}\,\sqrt{x} = \sqrt[6]{(x+y)^5}$ is $x^{1/2}\,y^{1/3} = (x+y)^{1/2 + 1/3}$, the homogeneous form $x^{m}y^{n} = (x+y)^{m+n}$ with $m=\tfrac12,\ n=\tfrac13$.

    Take logarithms: $m\log x + n\log y = (m+n)\log(x+y)$. Differentiate:

    $\dfrac{m}{x} + \dfrac{n}{y}\dfrac{dy}{dx} = \dfrac{(m+n)}{x+y}\left(1 + \dfrac{dy}{dx}\right).$

    Testing $\dfrac{dy}{dx} = \dfrac{y}{x}$: the right side becomes $\dfrac{(m+n)}{x+y}\cdot\dfrac{x+y}{x} = \dfrac{m+n}{x}$, which equals the left side $\dfrac{m}{x} + \dfrac{n}{x} = \dfrac{m+n}{x}$. Consistent, so

    $\dfrac{dy}{dx} = \dfrac{y}{x}.$

    ✓Final answer

    Option (D): $\dfrac{dy}{dx} = \dfrac{y}{x}$.