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, Problem 3RQ
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.

Expert Solution

Answer to Problem 3RQ

The statement is true.

Explanation of Solution

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.

Let y=g(x) and k=(v+g(x))h then substitute in equation (2),\

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

Consider f(g(x+h))f(g(x))

Substitute g(x+h)=g(x)+(v+g(x))h in the above equation,

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

Substitute f(g(x+h))f(g(x))=(w+f(g(x)))((v+g(x))h) in equation (1),

ddx(f(g(x)))=limh0(w+f(g(x)))((v+g(x))h)h=limh0(w+f(g(x)))(v+g(x))=limh0(wv+vf(g(x))+wg(x)+f(g(x))g(x))=f(g(x))g(x)           (Q limh0w=0 and limh0v=0)

Therefore, the given statement is true.

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