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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'.

22. y = sec(1 + x2)

To determine

To calculate: The derivative of y.

Explanation

Given:

The function is y=sec(1+x2).

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)

Calculation:

Obtain the derivative of y.

y=ddx(y)=ddx(sec(1+x2))

Let h(x)=1+x2 and f(u)=secu 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)            [u=h(x)]=ddu(f(u))=ddu(secu)=secutanu

Substitute u=1+x2 in the above equation,

f(h(x))

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