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Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343

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Section
BuyFindarrow_forward

Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343
Textbook Problem

Use the Chain Rule and the Product Rule to give an alternative proof of the Quotient Rule.

[Hint: Write f(x)/g(x) = f(x)[g(x)]–1.]

To determine

To find:  The derivative of f(x)g(x)=(f(x))(g(x))1.

Explanation

Result used: Chain Rule

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

F(x)=f(h(x))h(x) (1)

Derivative Rule:

(1) Power Rule: ddx(xn)=nxn1.

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

Calculation:

Obtain the derivative of f(x)g(x)=f(x)(g(x))1.

ddx(f(x)g(x))=ddx[(f(x))(g(x))1]

Apply the product rule (2),

ddx(f(x)g(x))=f(x)ddx[(g(x))1]+(g(x))1ddx(f(x))

=f(x)ddx[(g(x))1]+(g(x))1(f(x)) (2)

Obtain the derivative ddx[(g(x))1] using the chain rule as shown in equation (1).

Let h(x)=g(x) and f(u)=u1  where u=h(x).

ddx[(g(x))1]=f(h(x))h(x) (3)

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

f(h(x))=f(u)=ddu(f(u))=ddu(u1)

Apply derivative rule (1),

f(h(x))=(1)u11=u2=1u2

Substitute u=g(x) in the above equation,

f(h(x))=1(g(x))2

Thus, the derivative is f(h(x))=1(g(x))2

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