EXAMPLE 2Differentiate (a) y = sin(x³) and (b) sin (x).%3DSOLUTION(a) If y = sin(x), then the outer function is the sine function and the inner function is the power function, so the Chain Rule gives%3Ddydxdsin(x³)(x³)CoS=dxevaluatedat innerfunctionderivative evaluatedof outerfunctionderivativeof innerouterfunctionat innerfunctionfunction(b) Note that sin3(x) = (sin(x))'Here the outer function is the power function and the inner function is the sine function. Sod(sin(x)) 3dx(sin(x))23 ·%3Ddxinnerfunctionderivative of outerfunction evaluatedat inner functionderivativeof innerfunctionII

Given:-(a) y=sin(x3)(b)y=sin3xwe have to differentiate(a) y=sin(x3)differentiate with respect to…

EXAMPLE 2 Differentiate (a) y = sin(x³) and (b) sin3(x). SOLUTION (a) If y = sin(x³), then the outer function is the sine function and the inner function is the power function, so the Chain Rule gives dy dx d sin (x³) (x³) Cos = dx evaluated at inner function derivative evaluated derivative of inner function outer of outer function function at inner function (b) Note that sin3(x) = (sin(x))' Here the outer function is the power function and the inner function is the sine function. So dy dx d (sin(x)) 3 3· (sin(x))2 dx inner function derivative of outer function evaluated derivative of inner function at inner function

EXAMPLE 2 Differentiate (a) y = sin(x³) and (b) sin3(x). SOLUTION (a) If y = sin(x³), then the outer function is the sine function and the inner function is the power function, so the Chain Rule gives dy dx d sin (x³) (x³) Cos = dx evaluated at inner function derivative evaluated derivative of inner function outer of outer function function at inner function (b) Note that sin3(x) = (sin(x))' Here the outer function is the power function and the inner function is the sine function. So dy dx d (sin(x)) 3 3· (sin(x))2 dx inner function derivative of outer function evaluated derivative of inner function at inner function

Calculus: Early Transcendentals

8th Edition

ISBN:9781285741550

Author:James Stewart

Publisher:James Stewart

Chapter1: Functions And Models

Section: Chapter Questions

Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...

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