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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

Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement.

If f and g are differentiable, then

d d x [ f ( g ( x ) ) ] = f ' ( x ) g ' ( x )

To determine

Whether the statement, “If f and g are differentiable, then ddx[f(g(x))]=f(g(x))g(x)” is true or false.

Explanation

Definition used:

A function f(x) is differentiable at point x, then f(x)=limh0f(x+h)f(x)h.

Calculation:

By the definition of the derivative f(g(x)).

ddx(f(g(x)))=limh0f(g(x+h))f(g(x))h (1)

Since g is differentiable, then g(x)=limh0g(x+h)g(x)h

Let v=g(x+h)g(x)hg(x)

Thus,

v+g(x)=g(x+h)g(x)h(v+g(x))h=g(x+h)g(x)g(x+h)=g(x)+(v+g(x))h

Here, limh0v=0.

Since f is differentiable, then f(y)=limk0f(y+k)f(y)k

Let w=f(y+h)f(y)kf(y)

Thus,

w+f(y)=f(y+k)f(y)k(w+f(y))k=f(y+k)f(y)

f(y+k)=f(y)+(w+f(y))k (2)

Here, limk0w=0

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