   Chapter 2.3, Problem 102E

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# (a) If F ( x ) = f ( x ) g ( x ) where f and g have derivatives of all orders, show that. F ″ = f ″ g + 2 f ′ g ′ + f g ″ .(b) Find similar formulas for F ″ ′ and. F ( 4 ) .(c) Guess a formula for F ( n ) .

To determine

(a)

To prove:

F''=f''g+2 f'g'+f g''

Explanation

Concept used:

Product rule: If u and v are both differentiable, then

ddxuxvx=uxddxvx+vxddxux

Given:

Fx=fxgx. Also both function have derivatives of all order.

Proof Since both functions have derivatives of all order we may use product rule without any qualms.

Using the product rule

ddxFx= ddxfxgx=fxddxgx+gxddxfx

Differentiating once again we have

ddxddxFx=ddxddxfxgx

ddxddxfxgx=ddxfxddxgx+gxddxfx

d2dx2fxgx =ddxfxddxgx+ddxgxddxfx

For both parts on right side, apply the product rule again. So it becomes

d2dx2fxgx

=fxddxddxgx+ddxgx*ddx fx+gxddxddxfx+ ddxgx*ddx f(x<

To determine

(b)

To find:

Formula for F’’’ and F(4)

To determine

(c)

To guess:

Formula for F(n)

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