The derivative of tan^-1((2 x)/(1-x^2)) with respect to cos^-1 √(1-x^2) is (a) (√(1-x^2))/(1+x^2

Name: Problem 30, Chapter 3: fundamentals of mathematics differential calculus
Uploaded: 2022-02-26T16:10:02-08:00
Duration: 1 min
Channel: Raj Bala
Description: The derivative of tan^-1((2 x)/(1-x^2)) with respect to cos^-1 √(1-x^2) is (a) (√(1-x^2))/(1+x^2) (b) (1)/(√(1-x^2)) (c) (2)/(√(1-x^2)(1+x^2)) (d) (2 √(1-x^2))/(1+x^2)

step 1 Hello everyone, u is equal to 10 inverse under root 1 plus x square minus under root 1 minus x s

The derivative of tan^-1((2 x)/(1-x^2)) with respect to...

The derivative of $\tan ^{-1}\left(\frac{2 x}{1-x^{2}}\right)$ with respect to $\cos ^{-1} \sqrt{1-x^{2}}$ is
(a) $\frac{\sqrt{1-x^{2}}}{1+x^{2}}$
(b) $\frac{1}{\sqrt{1-x^{2}}}$
(c) $\frac{2}{\sqrt{1-x^{2}}\left(1+x^{2}\right)}$
(d) $\frac{2 \sqrt{1-x^{2}}}{1+x^{2}}$

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Key Concepts

Inverse Trigonometric Function Differentiation

This concept involves understanding the differentiation formulas and properties of inverse trigonometric functions such as tan?¹ and cos?¹. It requires knowing the standard derivative formulas and often involves applying the chain rule to functions composed with these inverses.

Chain Rule

The chain rule is a fundamental technique in calculus used to differentiate composite functions. When a function is composed of one function inside another, the derivative of the entire expression is the derivative of the outer function evaluated at the inner function multiplied by the derivative of the inner function.

Differentiation with Respect to Another Function

This concept refers to computing the derivative of one variable with respect to another when both variables are functions of a common independent variable. It is expressed as the ratio of their individual derivatives with respect to that common variable, enabling the computation of the derivative of one function indirectly.

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