   Chapter 2.5, Problem 70E

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# If g is a twice differentiable function and f ( x ) = x g ( x 2 ) , find f ′ ′ in terms of g , g ′ , and g ′ ′

To determine

To find:

f"  in terms of g, g and g

Explanation

Rules:

Chain rule:  Let Fx=fgx if g is differentiable at x and f is differentiable at g(x) then F'x=f'gxg'(x)

Product rule:

ddxfx.gx=f(x)ddxgx+g(x)ddxf(x)

Power rule: ddxxn=n.xn-1

Sum rule: ddxfx+gx=ddxfx+ddxg(x)

Constant multiplication rule: ddxkx=k.ddxx  where k is constant

Power function rule: ddxf(x)n=n.f(x)n-1*f'(x)

Given:

f(x)=xg(x2)

Calculation:

f(x)=xg(x2)

By using product rule,

f'x=ddxxgx2=x.ddxgx2+gx2.ddxx

By using chain rule,

f'x=x.g'x2.ddxx2+gx2.ddxx

By using power multiplication rule,

f'(x)=x.g'(x2)*2x+gx2*1

f'(x)=2x2.g'(x2)+gx2

Again differentiating f’ with respect to x,

f"(x)=ddxf'(x)=ddx2x2.g'(x2)+gx2

ddx2x2.g'(x2)+gx2=ddx2x2

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