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Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343

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BuyFindarrow_forward

Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343
Textbook Problem

Calculate y'.

20. y = exsec x

To determine

To calculate: The derivative of y.

Explanation

Given:

The function is y=exsecx.

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 rule:

Product rule: ddx(fg)=fddx(g)+gddx(f).

Calculation:

Obtain the derivative of y.

y=ddx(y)=ddx(exsecx)

Let h(x)=xsecx and f(u)=eu where u=h(x).

Apply the chain rule as shown in equation (1),

y=f(h(x))h(x) (2)

The derivative f(h(x)) is computed as follows,

f(h(x))=f(u)            [Qu=h(x)]=ddu(f(u))=ddu(eu)=eu

Substitute u=xsecx in the above equation,

f(h(x))=exsecx

Thus, the derivative f(h(x)) is f(h(x))=exsecx

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