BuyFind

Single Variable Calculus: Concepts...

4th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781337687805
BuyFind

Single Variable Calculus: Concepts...

4th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781337687805

Solutions

Chapter 3.4, Problem 3E
To determine

To find: The composite function in the form f(g(x)) and obtain the derivative of y.

Expert Solution

Answer to Problem 3E

The inner function is u=πx and the outer function is f(u)=tanu.

The derivative of y is y(x)=πsec2πx_.

Explanation of Solution

Given:

The function is y=tanπx.

Formula used:

The Chain Rule:

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

F(x)=g(h(x))h(x) (1)

Power Rule:

If n is positive integer, then ddx(xn)=nxn1 (2)

Calculation:

Let the inner function be u=g(x) and the outer function be y=f(u).

Then, g(x)=πx and f(u)=tanu. That is,

y=tanπx=f(πx)=f(g(x))

Therefore, y=f(g(x)).

Hence, the inner function is u=πx and the outer function is f(u)=tanu.

Thus, the required form of composite function is f(g(x))=tanu.

Obtain the derivative of y.

Let h(x)=πx and g(u)=tanu  where u=h(x).

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

y(x)=g(h(x))h(x) (3)

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

g(h(x))=g(u)=ddu(g(u))=ddu(tanu)=sec2u

Substitute u=πx in the above equation,

g(h(x))=sec2πx

Thus, the derivative g(h(x)) is g(h(x))=sec2πx.

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

h(x)=ddx(πx)

Apply the power rule as shown in equation (2),

h(x)=π[1x11]=π(1x0)=π(1)=π

Substitute sec2πx for g(h(x)) and π for h(x) in equation (3),

g(h(x))h(x)=sec2πx(π)=πsec2πx

Therefore, the derivative of y=tanπx is y(x)=πsec2πx_.

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