Differentiate tan^-1{(√(1+x^2)-√(1-x^2))/(√(1+x^2)+√(1-x^2))} w.r.t. cos^-1 x^2.

Name: Problem 109, Chapter 5: Differential Calculus Booster with Problems and Solutions for JEE Main and Advanced
Uploaded: 2022-03-07T11:46:31-08:00
Duration: 2 min 13 s
Channel: Rukhmani Jain
Description: Differentiate tan^-1{(√(1+x^2)-√(1-x^2))/(√(1+x^2)+√(1-x^2))} w.r.t. cos^-1 x^2.

step 1 In this question, let u is 10 inverse root 1 plus x squared minus root 1 minus x square upon roo

Differentiate...

Key Concepts

Differentiation with respect to a Function

Differentiating one function with respect to another can be understood through the relation df/dg = (df/dx)/(dg/dx), assuming both functions are differentiable with respect to an independent variable. This concept is critical when the problem requires finding the rate of change of one functional form in terms of another, linking their individual derivatives via the chain rule.

Quotient Rule

The quotient rule is used when differentiating a function that is the ratio of two differentiable functions. It provides a way to compute the derivative by taking into account the derivatives of both the numerator and the denominator. This rule becomes especially important when dealing with complex expressions arising from algebraic manipulations, such as those involving sums or differences of square roots in a quotient.

Inverse Trigonometric Functions

Inverse trigonometric functions, such as arctan and arccos, reverse the usual trigonometric relationships to yield an angle when given a ratio. Their differentiation relies on specific derivative formulas, like the derivative of arctan u being u'/(1+u²) and the derivative of arccos u being -u'/?(1-u²), and understanding these formulas is key to tackling problems that involve these functions in composite expressions.

Chain Rule

The chain rule is a fundamental concept in calculus used to differentiate composite functions, where one function is applied inside another. It states that the derivative of a composite function is the product of the derivative of the outer function evaluated at the inner function and the derivative of the inner function. When differentiating an expression with respect to another function, the chain rule systematically links the derivatives of the nested functions.

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