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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'.

49. y = cos ( e tan 3 x )

To determine

To calculate: The derivative of y.

Explanation

Given:

The function is y=cos(etan3x).

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 t and F is given by the product,

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

Calculation:

Obtain the derivative of f(t).

f(t)=ddt(f(t))=ddt(cos(etan3x))

Let h(x)=etan3x and g(u)=cosu  where u=h(x).

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

y=g(h(x))h(x) (2)

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

g(h(x))=f(u)=ddu(f(u))=ddu(cosu)=sinu

Substitute u=etan3x in the above equation,

g(h(x))=sin(etan3x)

Thus, the derivative is g(h(x))=sin(etan3x).

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

h(x)=ddx(etan3x)

Let k(x)=tan3x and s(u)=eu  where u=k(x).

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

h(x)=s(k(x))k(x) (3)

The derivative s(k(x)) is computed as follows,

s(k(x))=s(u)=ddu(s(u))=ddu(eu)=eu

Substitute u=tan3x in the above equation,

s(k(x))=etan3x

Thus, the derivative is s(k(x))=etan3x

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