AP EAMCET · Maths · Differentiation
The derivative of \(f(x)=x^{\tan ^{-1} x}\) with respect to \(g(x)=\sec ^{-1}\left(\frac{1}{2 x^2-1}\right)\) is
- A \(\frac{1}{2} \sqrt{1-x^2} x^{\tan ^{-1} x}\left[\frac{\log x}{1+x^2}+\frac{\tan ^{-1} x}{x}\right]\)
- B \(-\frac{1}{2} \sqrt{1-x^2} x^{x^{-a^{-1}}}\left[\log \left(\tan ^{-1} x\right)+x\left(1+x^2\right) \tan ^{-1} x\right]\)
- C \(\frac{\left.-2 \tan ^{-1} \frac{\log x}{1+x^2}+\frac{\tan ^{-1} x}{x}\right]}{\sqrt{1-x^2}}\)
- D \(-\frac{1}{2} \sqrt{1-x^2} x^{\tan x^{-1}}\left[\frac{\log x}{1+x^2}+\frac{\tan ^{-1} x}{x}\right]\)
Answer & Solution
Correct Answer
(D) \(-\frac{1}{2} \sqrt{1-x^2} x^{\tan x^{-1}}\left[\frac{\log x}{1+x^2}+\frac{\tan ^{-1} x}{x}\right]\)
Step-by-step Solution
Detailed explanation
We have, \[ \begin{aligned} f(x) & =x^{\tan ^{-1} x} \\ \Rightarrow \quad \log f(x) & =\tan ^{-1} x \log x \end{aligned} \]…
See the Complete Solution
Get step-by-step explanations for this and 2.5 Lakh+ more JEE, NEET & CET questions.
- Unlock all solutions
- Practice the full chapter
- Track accuracy across PYQs
4.8 rated on Google Play · 14,000+ reviews
More questions from Maths
- The amplitude of \(\sin \frac{\pi}{5}+i\left(1-\cos \frac{\pi}{5}\right)\) isAP EAMCET 2018 Medium
- Let \(\mathbf{u}\) and \(\mathbf{v}\) be two non-zero vectors. Then the magnitude of the cross product \(\mathbf{u} \times \mathbf{v}\) is alwaysAP EAMCET 2020 Easy
- \(\frac{d}{d x}\left(\sin ^{-1} \frac{3+4 x}{5 \sqrt{1+x^2}}\right)=\)AP EAMCET 2020 Easy
- To find the coefficient of \(x^4\) in the expansion of \(\frac{3 x}{(x-2)(x-1)}\), the interval in which the expansion isvalid, isAP EAMCET 2016 Medium
- If \((\sqrt{3}-\mathrm{i})^{\mathrm{n}}=2^{\mathrm{n}}, \mathrm{n} \in \mathrm{N}\), then the least possible value of n isAP EAMCET 2025 Medium
- When an unfair dice is thrown, the probability of getting a number \(k\) on it is \(\mathrm{P}(\mathrm{X}=k)=k^2 \mathrm{P}\), where \(k=1,2,3,4,5,6\) and X is the random variable denoting a number on the dice, then the mean of X isAP EAMCET 2024 Easy
More PYQs from AP EAMCET
- In a hydraulic lift, if the radius of the smaller piston is 5 cm, and the radius of the larger piston is 50 cm, then the weight that the larger piston can support when a force of 250 N is applied to the smaller piston isAP EAMCET 2025 Easy
- Three charges \(4 q, Q\) and \(q\) are placed at positions \(0, \frac{l}{2}\) and \(l\) respectively along a straight line. If the resultant force on \(q\) is zero, then \(Q\) is equal toAP EAMCET 2020 Medium
- Identify the correct statements from the following.
(A) The ionic product of water at \(100^{\circ} \mathrm{C}\) is \( < 10^{-14}\).
(B) \(\mathrm{pH}\) of a solution decreases with increasing temperature.
(C) \(\mathrm{NaH}_2 \mathrm{PO}_4\) on hydrolysis gives a basic solution.
(D) \(\mathrm{NH}_3\) can act both as Bronsted acid and base.AP EAMCET 2022 Medium - Lenz's law is consequence of the law of conservation ofAP EAMCET 2020 Easy
- A light of intensity incidents on a black surface of area . The radiation pressure on the surface isAP EAMCET 2022 Medium
- \(\int \frac{\sec ^2 x}{(\sec x+\tan x)^{5 / 2}} d x=\)AP EAMCET 2025 Medium