The function is defined for

MATHEMATICS · Multi-part · CBSE · Mathematics · Continuity and Differentiability · Continuity and Differentiability

Problem

The function y=arcsin⁡xy=\arcsin x is defined for −1<x<1-1<x<1. In this problem you will establish the derivative formula for arcsin⁡x\arcsin x from first principles and determine its domain of validity. (a) Rewrite the statement y=arcsin⁡xy=\arcsin x as an equivalent relation that does not involve any inverse trigonometric notation. (b) Differentiate the relation obtained in part (a) with respect to xx to obtain an expression for dydx\dfrac{dy}{dx} that may still involve yy. Show each step clearly. (c) Express dydx\dfrac{dy}{dx} entirely in terms of xx. Explain how the sign of cos⁡y\cos y is determined on the interval −1<x<1-1<x<1 and justify the final sign appearing in your derivative. (d) State the final derivative formula for ddx[arcsin⁡x]\dfrac{d}{dx}[\arcsin x] together with the open interval on which the formula is valid.

Related theory

Browse the problem library