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Calculus (MindTap Course List)

8th Edition
James Stewart
ISBN: 9781285740621

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Calculus (MindTap Course List)

8th Edition
James Stewart
ISBN: 9781285740621
Textbook Problem
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Find the first and second derivatives of the function.

f ( x ) = x 2 1 + 2 x

To determine

To find: First and second derivative of the function.

Explanation

1) Concept:

By differentiating the given function we can obtain first derivative. To find second derivative we shall differentiate fist derivative. We may use rules of differentiation to compute derivatives.

2) Formula:

Let f and g are differentiable functions then

i. Quotient rule: ddxfxgx=gx.ddxfx-fxddx[gx]gx2

ii. Addition rule: ddxfx+gx=ddxfx+ddxg(x)

iii. Power rule:  ddxxn=nxn-1

iv. Derivative of constant multiple: ddx kfx=kddxfx

v. Derivative of constant: ddxk=0 where k is constant.

3) Given:

fx=x21+2x

4) Calculations:

Differentiating given function with respect to x,

By using quotient rule,

f'(x)=ddxx21+2x=(1+2x).ddxx2-x2ddx[1+2x]1+2x2

By using addition rule,

f'(x)=1+2x.ddxx2-x2ddx1+ddx2x1+2x2

By using power and constant rule,

f'(x)= 1+2x2x-x2(0+2)1+2x2

         = 2x+4x2-2x21+2x2 Simplifying,

f'x= 2x+2x21+2x2

 To find second derivative differentiate f(x) with respect to x,

By using quotient rule,

f"(x) = ddx2x+2x21+2x2

            =1+2x2

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