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Calculus: Early Transcendentals

8th Edition
James Stewart
ISBN: 9781285741550

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BuyFindarrow_forward

Calculus: Early Transcendentals

8th Edition
James Stewart
ISBN: 9781285741550
Textbook Problem

Calculate y'.

11. y = x cos x

To determine

To find: The derivative of y.

Explanation

Given:

The function is y=xcosx.

Result used: Chain Rule

If g is differentiable at t and f is differentiable at g(t), then the composite function F=fg defined by F(t)=f(g(t)) is differentiable at x and F is given by the product

F(t)=f(g(t))g(t) (1)

Derivative Rules:

Product Rule: ddx[f(x)g(x)]=f(x)ddx[g(x)]+g(x)ddx[f(x)]

Calculation:

Obtain the derivative of y.

y=ddx(y)=ddx(xcosx)

Apply the product rule,

y=xddx(cosx)+(cosx)ddx(x)=xddx(cosx)+(cosx)ddx(x12)=xddx(cosx)+(cosx)(12x121)

=xddx(cosx)+(cosx)(12x12) (2)

Obtain the derivative ddx(cosx) by using the chain rule as shown in equation (1),

Let h(x)=x and f(u)=cosu  where u=h(x)

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