Let f(x)=sin^-1(sinx), ∀x ∈[-π, 2 π]. Then find f^'(x). |...

Name: Problem 152, Chapter 2: Differential Calculus Booster with Problems and Solutions for JEE Main and Advanced
Uploaded: 2022-04-07T13:34:09-08:00
Duration: 41 s
Channel: Rukhmani Jain
Description: Let f(x)=sin^-1(sinx), ∀x ∈[-π, 2 π]. Then find f^'(x).

Key Concepts

Inverse Sine Function

The inverse sine function, often written as arcsin or sin?¹, is defined such that for any value within its domain [-1, 1], it returns the unique angle in the interval [-?/2, ?/2] whose sine equals the original value. This concept is fundamental in problems involving the composition of functions, as it explains why sin?¹(sin x) does not simply equal x for all values of x but must be adjusted to lie within the principal range.

Function Composition with Inverse Trigonometric Functions

When a trigonometric function is composed with its inverse, as in sin?¹(sin x), special attention must be paid to the fact that the inverse function only retrieves angles within a limited interval (the principal value range). This requires reformulation of the expression for different intervals of x, a key idea in ensuring that the composite function accurately reflects the periodic nature of the sine function.

Piecewise Function Analysis

Since sin?¹(sin x) does not equal x for every x due to the restricted range of the inverse sine function, the function must be described piecewise over different intervals. Each piece corresponds to an interval where the relationship between x and sin?¹(sin x) follows a specific linear rule. Understanding and formulating these piecewise segments is crucial in both expressing the function correctly and differentiating it.

Differentiation of Piecewise Functions

Differentiating a function defined in a piecewise manner involves computing the derivative for each segment separately, while also being mindful of the points of transition between the pieces. This approach is essential for finding the derivative of sin?¹(sin x), particularly because each piece may yield a different derivative expression based on its linear form and the underlying rules of differentiation.

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