Q.Find the value of the following: $\tan \frac{1}{2}\left[\sin^{-1} \frac{2x}{1+x^2} + \cos^{-1} \frac{1-y^2}{1+y^2}\right]$, where $|x| < 1$, $y > 0$ and $xy < 1$.

Showing the 12 most recent of 34 on this concept.

1. AP EAPCET 2021Set eng-2021-08-19-FN1 markMCQ
   Q.If $y = \tan^{-1}\left(\dfrac{\sqrt{1+x^2} + \sqrt{1-x^2}}{\sqrt{1+x^2} - \sqrt{1-x^2}}\right)$, where $x^2 \le 1$. Then find $\dfrac{dy}{dx}$ (A) $\dfrac{\pi}{4} + \dfrac{1}{2}\cos^{-1}(x^2)$ (B) $\dfrac{\pi}{4} - \dfrac{1}{2}\cos^{-1}(x^2)$ (C) $\dfrac{-x}{\sqrt{1-x^4}}$ (D) $\dfrac{-2x}{\sqrt{1-x^4}}$

   ›Reveal solutionSolution

   Substituting $x^2=\cos\varphi$ collapses the expression to $y=\pi/4+\tfrac12\cos^{-1}(x^2)$, whose derivative is $\dfrac{-x}{\sqrt{1-x^4}}$.

   Concept and Intuition

   Nested-radical inverse-trig expressions like this are almost always designed to simplify via a trig substitution that turns $\sqrt{1\pm x^2}$ into $\sqrt2\cos$ or $\sqrt2\sin$ of a half-angle, converting the whole ratio into a single tangent — much easier to differentiate than the raw radical expression.

   Step-by-Step Solution

   1. Let $x^2=\cos\varphi$ (valid since $x^2\le1$). Then $1+x^2=2\cos^2(\varphi/2)$ and $1-x^2=2\sin^2(\varphi/2)$.
   2. So $\sqrt{1+x^2}=\sqrt2\cos(\varphi/2)$ and $\sqrt{1-x^2}=\sqrt2\sin(\varphi/2)$.
   3. The ratio becomes $\dfrac{\cos(\varphi/2)+\sin(\varphi/2)}{\cos(\varphi/2)-\sin(\varphi/2)} = \dfrac{1+\tan(\varphi/2)}{1-\tan(\varphi/2)} = \tan\!\left(\dfrac{\pi}{4}+\dfrac{\varphi}{2}\right)$.
   4. So $y=\tan^{-1}\!\left[\tan\!\left(\dfrac\pi4+\dfrac\varphi2\right)\right]=\dfrac\pi4+\dfrac\varphi2=\dfrac\pi4+\dfrac12\cos^{-1}(x^2)$. …

   🔒 Start your 14-day free trial to see the full solution →
2. AP EAPCET 2024Set eng-2024-05-22-AN1 markMCQ
   Q.If $y=\text{Tan}^{-1}\dfrac{x}{1+2x^2}+\text{Tan}^{-1}\dfrac{x}{1+6x^2}+\text{Tan}^{-1}\dfrac{x}{1+12x^2}$, then $\left(\dfrac{dy}{dx}\right)_{x=\frac{1}{2}}=$ (A) $1$ (B) $-1$ (C) $0$ (D) $\frac{1}{2}$

   ›Reveal solutionSolution

   Recognize the telescoping arctan pattern to collapse $y$ to $\arctan(4x)-\arctan(x)$; the derivative at $x=\frac12$ is $0$.

   Concept and Intuition

   The identity $\arctan A-\arctan B=\arctan\dfrac{A-B}{1+AB}$ (mod branch issues) means a sum like $\arctan\dfrac{x}{1+n(n+1)x^2}$, recognized as $\arctan((n+1)x)-\arctan(nx)$, telescopes when summed over consecutive $n$ — a huge simplification before ever differentiating.

   Step-by-Step Solution

   1. Check the general term: $\arctan((n+1)x)-\arctan(nx)=\arctan\dfrac{(n+1)x-nx}{1+n(n+1)x^2}=\arctan\dfrac{x}{1+n(n+1)x^2}$.
   2. Match given terms: $\dfrac{x}{1+2x^2}$ has $n(n+1)=2\Rightarrow n=1$; $\dfrac{x}{1+6x^2}$ has $n(n+1)=6\Rightarrow n=2$; $\dfrac{x}{1+12x^2}$ has $n(n+1)=12\Rightarrow n=3$.
   3. So $y=[\arctan2x-\arctan x]+[\arctan3x-\arctan2x]+[\arctan4x-\arctan3x]=\arctan4x-\arctan x$ (everything else cancels). …

   🔒 Start your 14-day free trial to see the full solution →
3. AP EAPCET 2025Set eng-2025-05-22-AN1 markMCQ
   Q.If $\dfrac{1}{2}\sin^{-1}\left(\dfrac{3\sin 2\theta}{5+4\cos 2\theta}\right) = \tan^{-1}x$ then $x =$ (A) $\tan\dfrac{\theta}{3}$ (B) $\dfrac{1}{3}\tan\theta$ (C) $\tan 3\theta$ (D) $\dfrac{1}{3}\tan 3\theta$

   ›Reveal solutionSolution

   Rewrite the inverse-sine argument as $\sin(2\phi)$ for a suitable $\tan\phi$ built from $\tan\theta$, so the half of the inverse sine collapses to $\tan^{-1}$ of that quantity directly. Answer: $x=\dfrac13\tan\theta$.

   Concept and Intuition

   Expressions of the form $\dfrac{k\sin2\theta}{A+B\cos2\theta}$ can often be massaged, after dividing through by $(1+\tan^2\theta)$ i.e. converting to $t=\tan\theta$, into the recognizable double-angle sine pattern $\dfrac{2u}{1+u^2}=\sin(2\tan^{-1}u)$ for some rescaled variable $u$. Spotting this pattern converts an awkward inverse-trig expression into a clean angle.

   Step-by-Step Solution

   1. Let $t=\tan\theta$. Recall $\sin2\theta=\dfrac{2t}{1+t^2}$, $\cos2\theta=\dfrac{1-t^2}{1+t^2}$.
   2. Numerator: $3\sin2\theta=\dfrac{6t}{1+t^2}$.
   3. Denominator: $5+4\cos2\theta=5+\dfrac{4(1-t^2)}{1+t^2}=\dfrac{5(1+t^2)+4(1-t^2)}{1+t^2}=\dfrac{9+t^2}{1+t^2}$.
   4. So the ratio is $\dfrac{6t/(1+t^2)}{(9+t^2)/(1+t^2)}=\dfrac{6t}{9+t^2}$.
   5. Let $u=t/3$. Then $\dfrac{2u}{1+u^2}=\dfrac{2t/3}{1+t^2/9}=\dfrac{2t/3}{(9+t^2)/9}=\dfrac{6t}{9+t^2}$ — exactly matches!
   6. So $\dfrac{6t}{9+t^2}=\sin(2\phi)$ where $\tan\phi=u=t/3$, i.e. $\phi=\tan^{-1}\!\left(\dfrac{\tan\theta}{3}\right)$. …

   🔒 Start your 14-day free trial to see the full solution →
4. AP EAPCET 2021Set eng-2021-08-24-AN1 markMCQ
   Q.If $y=\text{Tan}^{-1}\sqrt{\dfrac{1-\cos x}{1+\cos x}}$, then the values of $\dfrac{dy}{dx}$ and $\dfrac{d^2y}{dx^2}$ respectively are ________ (A) $1,\ 0$ (B) $\dfrac{x}{2},\ \dfrac{1}{2}$ (C) $\dfrac{1}{2},\ 0$ (D) $\dfrac{-1}{2},\ 0$

   ›Reveal solutionSolution

   The half-angle identity collapses $y$ to $x/2$, so $dy/dx = 1/2$ and $d^2y/dx^2 = 0$.

   Concept and Intuition

   Expressions like $\sqrt{\dfrac{1-\cos x}{1+\cos x}}$ are classic half-angle simplifications; recognizing the identity turns an intimidating inverse-trig derivative problem into a trivial linear function.

   Step-by-Step Solution

   1. Recall $1-\cos x = 2\sin^2(x/2)$ and $1+\cos x = 2\cos^2(x/2)$.
   2. So $\dfrac{1-\cos x}{1+\cos x} = \tan^2(x/2)$, and $\sqrt{\tan^2(x/2)} = |\tan(x/2)|$.
   3. On the principal branch where $\tan(x/2)\ge 0$ (i.e. $x\in(-\pi,\pi)$), $y = \operatorname{Tan}^{-1}(\tan(x/2)) = \dfrac{x}{2}$.
   4. Differentiate: $\dfrac{dy}{dx} = \dfrac12$.
   5. Differentiate again: since $dy/dx$ is constant, $\dfrac{d^2y}{dx^2} = 0$.

   Common Mistakes …

   🔒 Start your 14-day free trial to see the full solution →
5. AP EAPCET 2026Set eng-2026-05-13-AN1 markMCQ
   Q.$\sin\left(\text{Tan}^{-1}\dfrac{12}{17} + \text{Tan}^{-1}\dfrac{5}{29}\right) =$ (A) $1$ (B) $\dfrac{1}{2}$ (C) $\dfrac{\sqrt{3}}{2}$ (D) $\dfrac{1}{\sqrt{2}}$

   ›Reveal solutionSolution

   This tests the sine-addition formula built from two inverse-tangent right triangles; the arithmetic collapses very neatly since $866=2\times433$.

   Concept and Intuition

   $\tan^{-1}\frac{12}{17}$ is the angle of a right triangle with opposite $12$, adjacent $17$, hypotenuse $\sqrt{12^2+17^2}=\sqrt{433}$. Similarly $\tan^{-1}\frac{5}{29}$ gives a triangle with hypotenuse $\sqrt{5^2+29^2}=\sqrt{866}$. Once we have $\sin$ and $\cos$ of both angles, the sum formula does the rest — no need to ever find the angles themselves.

   Step-by-Step Solution

   1. Let $A=\tan^{-1}\frac{12}{17}$: $\sin A=\dfrac{12}{\sqrt{433}}$, $\cos A=\dfrac{17}{\sqrt{433}}$ (since $12^2+17^2=144+289=433$).
   2. Let $B=\tan^{-1}\frac{5}{29}$: $\sin B=\dfrac{5}{\sqrt{866}}$, $\cos B=\dfrac{29}{\sqrt{866}}$ (since $5^2+29^2=25+841=866$).
   3. $\sin(A+B)=\sin A\cos B+\cos A\sin B=\dfrac{12(29)+17(5)}{\sqrt{433}\sqrt{866}}=\dfrac{348+85}{\sqrt{433\cdot866}}=\dfrac{433}{\sqrt{433\cdot866}}$. …

   🔒 Start your 14-day free trial to see the full solution →
6. AP EAPCET 2025Set eng-2025-05-23-AN1 markMCQ
   Q.The number of solutions of $\operatorname{Tan}^{-1}1+\dfrac{1}{2}\operatorname{Cos}^{-1}x^2-\operatorname{Tan}^{-1}\left(\dfrac{\sqrt{1+x^2}+\sqrt{1-x^2}}{\sqrt{1+x^2}-\sqrt{1-x^2}}\right)=0$ is (A) 3 (B) 0 (C) 1 (D) infinitely many

   ›Reveal solutionSolution

   This tests simplifying an inverse-trig expression via a trigonometric substitution, revealing that the equation is actually an identity over its whole domain. Answer: infinitely many solutions.

   Concept and Intuition

   The fraction $\dfrac{\sqrt{1+x^2}+\sqrt{1-x^2}}{\sqrt{1+x^2}-\sqrt{1-x^2}}$ looks intimidating, but substituting $x^2=\cos\alpha$ converts $1\pm x^2$ into $2\cos^2(\alpha/2)$ and $2\sin^2(\alpha/2)$, turning the whole fraction into a clean $\tan\left(\frac{\pi}{4}+\frac{\alpha}{2}\right)$. This is the standard trick for expressions of the form $\sqrt{1+x^2}\pm\sqrt{1-x^2}$.

   Step-by-Step Solution

   1. Domain: $\operatorname{Cos}^{-1}(x^2)$ needs $x^2\in[-1,1]$, and since $x^2\ge0$ always, effectively $x^2\in[0,1]$, i.e. $x\in[-1,1]$. Also need $\sqrt{1-x^2}$ real, consistent.
   2. Let $\alpha=\operatorname{Cos}^{-1}(x^2)\in[0,\pi/2]$ (since $x^2\in[0,1]$, $\alpha$ can only range over $[0,\pi/2]$, not the full $[0,\pi]$).
   3. Then $x^2=\cos\alpha$, so $1+x^2=1+\cos\alpha=2\cos^2(\alpha/2)$ and $1-x^2=1-\cos\alpha=2\sin^2(\alpha/2)$. Since $\alpha/2\in[0,\pi/4]$, both $\cos(\alpha/2),\sin(\alpha/2)\ge0$, so $\sqrt{1+x^2}=\sqrt2\cos(\alpha/2)$, $\sqrt{1-x^2}=\sqrt2\sin(\alpha/2)$.
   4. The fraction becomes $\dfrac{\cos(\alpha/2)+\sin(\alpha/2)}{\cos(\alpha/2)-\sin(\alpha/2)} = \dfrac{1+\tan(\alpha/2)}{1-\tan(\alpha/2)} = \tan\left(\dfrac{\pi}{4}+\dfrac{\alpha}{2}\right)$.
   5. Since $\alpha/2\in[0,\pi/4]$, we have $\dfrac{\pi}{4}+\dfrac{\alpha}{2}\in\left[\dfrac{\pi}{4},\dfrac{\pi}{2}\right)$, safely inside the principal range of $\operatorname{Tan}^{-1}$, so $\operatorname{Tan}^{-1}\left[\tan\left(\frac{\pi}{4}+\frac{\alpha}{2}\right)\right]=\dfrac{\pi}{4}+\dfrac{\alpha}{2}$ exactly (excluding $x=0$ where $\alpha=\pi/2$ makes the denominator zero). …

   🔒 Start your 14-day free trial to see the full solution →
7. AP EAPCET 2026Set eng-2026-05-15-AN1 markMCQ
   Q.$\sin\left[2\tan^{-1}\left(\dfrac{1}{2}\right) + \sin^{-1}\left(\dfrac{3}{5}\right)\right] =$ (A) 0 (B) 1 (C) $\dfrac{1}{2}$ (D) $\dfrac{\sqrt3}{2}$

   ›Reveal solutionSolution

   Convert both inverse-trig terms into a single angle's sine/cosine using right-triangle ratios, then apply the sine addition formula; the sum evaluates neatly to $1$.

   Concept and Intuition

   When adding two inverse trig angles, the cleanest approach is to name each one, extract its sine and cosine from the implied right triangle, then use the standard addition formula $\sin(A+B)=\sin A\cos B+\cos A\sin B$ — never try to add the angles numerically.

   Step-by-Step Solution

   1. Let $\varphi = \tan^{-1}\left(\dfrac12\right)$. In a right triangle, opposite $=1$, adjacent $=2$, hypotenuse $=\sqrt5$.
   2. Use the double-angle identity $\tan2\varphi = \dfrac{2\tan\varphi}{1-\tan^2\varphi} = \dfrac{2(1/2)}{1-(1/4)} = \dfrac{1}{3/4}=\dfrac43$.
   3. Since $\varphi\in(0,\pi/4)$ (as $\tan\varphi=1/2<1$), $2\varphi\in(0,\pi/2)$, so this is a genuine first-quadrant angle: with opposite $4$, adjacent $3$, hypotenuse $5$, giving $\sin2\varphi=4/5,\ \cos2\varphi=3/5$.
   4. Let $\psi=\sin^{-1}(3/5)$, so $\sin\psi=3/5$ and (principal branch, first quadrant) $\cos\psi=4/5$. …

   🔒 Start your 14-day free trial to see the full solution →
8. AP EAPCET 2025Set eng-2025-05-24-FN1 markMCQ
   Q.For $0 < x < 1$, $\displaystyle\int [\mathrm{Tan}^{-1}(1 - x + x^2) + \mathrm{Tan}^{-1}(1 - x)]\, dx =$ (A) $x\,\mathrm{Cot}^{-1}x + \log\sqrt{1+x^2} + c$ (B) $x\,\mathrm{Tan}^{-1}x - \log(1+x^2) + c$ (C) $x\,\mathrm{Cot}^{-1}x + \dfrac{3}{4}\log(1+x^2) + c$ (D) $x\,\mathrm{Tan}^{-1}x - \dfrac{3}{4}\log\sqrt{1+x^2} + c$

   ›Reveal solutionSolution

   The two arctangent terms combine, via the tan-addition identity, into a single $\mathrm{Cot}^{-1}x$; integrating that by parts gives option (A).

   Concept and Intuition

   The stem looks intimidating because it has two separate inverse-tangent terms with messy arguments. The key insight is that $\mathrm{Tan}^{-1}p+\mathrm{Tan}^{-1}q$ always collapses via

   $\mathrm{Tan}^{-1}p+\mathrm{Tan}^{-1}q=\mathrm{Tan}^{-1}\!\left(\frac{p+q}{1-pq}\right)\ (\text{mod }\pi\text{ correction}),$

   so it's worth testing whether $p=1-x+x^2$ and $q=1-x$ are designed to make $\dfrac{p+q}{1-pq}$ simplify beautifully — which they are.

   Step-by-Step Solution

   1. Compute $p+q = (1-x+x^2)+(1-x) = 2-2x+x^2$.
   2. Compute $pq = (1-x+x^2)(1-x)$. Expanding: $(1-x+x^2)(1-x) = 1-2x+2x^2-x^3$.
   3. So $1-pq = 1-(1-2x+2x^2-x^3) = 2x-2x^2+x^3 = x(2-2x+x^2)$.
   4. Hence $\dfrac{p+q}{1-pq} = \dfrac{2-2x+x^2}{x(2-2x+x^2)} = \dfrac{1}{x}$.
   5. Check the correction term: for $0<x<1$, $2-2x+x^2=(x-1)^2+1>0$ and $x>0$, so $1-pq>0 \Rightarrow pq<1$, meaning the plain addition formula applies with no $\pm\pi$ shift.
   6. So the integrand is exactly $\mathrm{Tan}^{-1}(1/x) = \mathrm{Cot}^{-1}x$ (valid since $x>0$).
   7. Now integrate by parts: $\displaystyle\int \mathrm{Cot}^{-1}x\,dx = x\,\mathrm{Cot}^{-1}x - \int x\cdot\left(\frac{-1}{1+x^2}\right)dx = x\,\mathrm{Cot}^{-1}x+\int\frac{x}{1+x^2}dx$.
   8. $\displaystyle\int\frac{x}{1+x^2}dx = \frac12\log(1+x^2)=\log\sqrt{1+x^2}$.
   9. Total: $x\,\mathrm{Cot}^{-1}x+\log\sqrt{1+x^2}+c$. …

   🔒 Start your 14-day free trial to see the full solution →
9. AP EAPCET 2021Set eng-2021-08-19-AN1 markMCQ
   Q.$\dfrac{d}{dx}\left(\operatorname{Tan}^{-1}\left(\dfrac{\cos x}{1+\sin x}\right)\right)=$ (A) $\dfrac{1}{2}$ (B) $\dfrac{-1}{2}$ (C) $1$ (D) $-1$

   ›Reveal solutionSolution

   Simplify the argument to a single tangent of a half-angle expression, then differentiate. Answer: $-\dfrac12$.

   Concept and Intuition

   Expressions like $\dfrac{\cos x}{1+\sin x}$ are classic half-angle simplifications that collapse to $\tan\left(\dfrac\pi4-\dfrac x2\right)$, letting the inverse tangent cancel with the tangent directly.

   Step-by-Step Solution

   1. Multiply numerator and denominator by $(1-\sin x)$: $\dfrac{\cos x}{1+\sin x}=\dfrac{\cos x(1-\sin x)}{1-\sin^2x}=\dfrac{\cos x(1-\sin x)}{\cos^2x}=\dfrac{1-\sin x}{\cos x}$.
   2. This is a known identity: $\dfrac{1-\sin x}{\cos x}=\tan\left(\dfrac\pi4-\dfrac x2\right)$. …

   🔒 Start your 14-day free trial to see the full solution →
10. AP EAPCET 2025Set eng-2025-05-22-AN1 markMCQ
    Q.If $y = \mathrm{Tan}^{-1}\left(\dfrac{x}{1+2x^2}\right) + \mathrm{Tan}^{-1}\left(\dfrac{x}{1+6x^2}\right)$, then $\dfrac{dy}{dx}=$ (A) $\dfrac{4}{16x^2+1} - \dfrac{3}{9x^2+1}$ (B) $\dfrac{3}{9x^2+1} - \dfrac{1}{x^2+1}$ (C) $\dfrac{3}{9x^2+1} - \dfrac{2}{4x^2+1}$ (D) $\dfrac{1}{9x^2+1} - \dfrac{1}{x^2+1}$

    ›Reveal solutionSolution

    Recognizing each arctan term as a telescoping difference $\tan^{-1}(2x)-\tan^{-1}(x)$ and $\tan^{-1}(3x)-\tan^{-1}(2x)$ collapses $y$ to $\tan^{-1}3x-\tan^{-1}x$, whose derivative is immediate.

    Concept and Intuition

    Terms of the form $\tan^{-1}\!\left(\dfrac{a-b}{1+ab}\right)$ are exactly $\tan^{-1}a - \tan^{-1}b$ (the tangent subtraction identity, valid when $ab>-1$). Spotting this pattern turns an awkward-looking sum into a telescoping simplification, avoiding messy direct differentiation of nested rational-argument arctans.

    Step-by-Step Solution

    1. Compare $\dfrac{x}{1+2x^2}$ to the form $\dfrac{a-b}{1+ab}$: try $a=2x,\,b=x$, giving $\dfrac{2x-x}{1+2x\cdot x}=\dfrac{x}{1+2x^2}$ ✓. So

    $\tan^{-1}\!\left(\frac{x}{1+2x^2}\right) = \tan^{-1}(2x)-\tan^{-1}(x)$

    2. Compare $\dfrac{x}{1+6x^2}$ similarly: try $a=3x,\,b=2x$, giving $\dfrac{3x-2x}{1+3x\cdot2x}=\dfrac{x}{1+6x^2}$ ✓. So

    $\tan^{-1}\!\left(\frac{x}{1+6x^2}\right) = \tan^{-1}(3x)-\tan^{-1}(2x)$

    3. Add the two:

    $y = \big[\tan^{-1}2x-\tan^{-1}x\big] + \big[\tan^{-1}3x-\tan^{-1}2x\big] = \tan^{-1}3x - \tan^{-1}x$

    (the $\tan^{-1}2x$ terms cancel — a telescoping sum).

    4. Differentiate: …

    🔒 Start your 14-day free trial to see the full solution →
11. AP EAPCET 2021Set eng-2021-08-20-AN1 markMCQ
    Q.If $y = \log_{\cot x} \tan x - \log_{\tan x} \cot x + \tan^{-1}\left(\dfrac{4x}{4-x^2}\right)$, then $\dfrac{dy}{dx} =$ ______ (A) $\dfrac{1}{4+x^2}$ (B) $\dfrac{4}{4+x^2}$ (C) $\dfrac{1}{4-x^2}$ (D) $\dfrac{4}{4-x^2}$

    ›Reveal solutionSolution

    This tests the change-of-base log identity and the double-angle form of $\tan^{-1}$. The two log terms cancel completely, and the answer is $\dfrac{4}{4+x^2}$.

    Concept and Intuition

    Whenever you see $\log_a b$ and $\log_b a$ together, remember they are reciprocals of each other: $\log_a b=1/\log_b a$. Here $a=\cot x,\,b=\tan x$ are reciprocals of each other too, so $\ln(\tan x)=-\ln(\cot x)$, which forces both log terms to equal $-1$ and cancel. What's left is the classic $\tan^{-1}\left(\dfrac{2t}{1-t^2}\right)=2\tan^{-1}t$ substitution pattern with $t=x/2$.

    Step-by-Step Solution

    1. $\log_{\cot x}\tan x=\dfrac{\ln\tan x}{\ln\cot x}=\dfrac{\ln\tan x}{\ln(1/\tan x)}=\dfrac{\ln\tan x}{-\ln\tan x}=-1$.
    2. Similarly $\log_{\tan x}\cot x=-1$.
    3. So $y=-1-(-1)+\tan^{-1}\!\left(\dfrac{4x}{4-x^2}\right)=\tan^{-1}\!\left(\dfrac{4x}{4-x^2}\right)$. …

    🔒 Start your 14-day free trial to see the full solution →
12. AP EAPCET 2021Set eng-2021-08-20-FN1 markMCQ
    Q.If $y = \mathrm{Tan}^{-1}\left\{\dfrac{ax-b}{bx+a}\right\}$, then $y' =$ _______ (A) $\dfrac{1}{1+x^2} + \dfrac{a^2}{a^2+b^2}$ (B) $\dfrac{1}{1+x^2}$ (C) $\dfrac{1}{1+\left(\dfrac{ax-b}{bx+a}\right)^2}$ (D) $\dfrac{bx+a}{1+(ax-b)^2}$

    ›Reveal solutionSolution

    This tests differentiating an arctangent of a Möbius-type expression, which (as often happens) collapses to the simple form $\tfrac{1}{1+x^2}$. Answer: $\dfrac{1}{1+x^2}$.

    Concept and Intuition

    Expressions of the form $\mathrm{Tan}^{-1}\!\left(\dfrac{ax-b}{bx+a}\right)$ often equal $\mathrm{Tan}^{-1}(x) - \mathrm{Tan}^{-1}(b/a)$ up to a constant (since $\dfrac{ax-b}{bx+a} = \dfrac{\frac{a}{b}x - 1}{x+\frac ab}$ resembles $\tan(\alpha-\beta)$ with a constant angle $\beta$), so its derivative should reduce to just $\dfrac{1}{1+x^2}$ — the constant angle contributes zero derivative.

    Step-by-Step Solution

    1. Let $u = \dfrac{ax-b}{bx+a}$. By the quotient rule, $u' = \dfrac{a(bx+a) - (ax-b)b}{(bx+a)^2} = \dfrac{abx+a^2-abx+b^2}{(bx+a)^2} = \dfrac{a^2+b^2}{(bx+a)^2}$.
    2. Compute $1+u^2 = 1+\dfrac{(ax-b)^2}{(bx+a)^2} = \dfrac{(bx+a)^2+(ax-b)^2}{(bx+a)^2}$. …

    🔒 Start your 14-day free trial to see the full solution →