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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 is differentiable, then d d x f ( x ) = f ' ( x ) 2 f ( x ) .

To determine

Whether the statement, “If f is differentiable, then ddx[f(x)]=f(x)2f(x)” is true or false.

Explanation

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 x and F is given by the product

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

Calculation:

Consider ddx[f(x)]=f(x)2f(x).

Take the left hand side ddx[f(x)].

ddx[f(x)]=ddx[f(x)]12

Let h(x)=f(x) and g(u)=u12 where u=h(x).

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

ddx[f(x)]=g(h(x))h(x) (2)

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

g(h(x))=g(u)            [Qu=h(x)]=ddu(g(u))=ddu(u12)=<

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