   Chapter 2.5, Problem 90E

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# If y = f ( u ) and u = g ( x ) , where f and g possess third derivatives, find a formula for d 3 y / d x 3 similar to the one given in Exercise 89.

To determine

To find:

Formula for d3ydx3

Explanation

We shall use the answer from exercise 89. Differentiate it one more time using chain rule.

Formula used:

Product rule:

ddxfxgx=f'xgx+fxg'(x)

Chain rule:

ddx(fgx=f'gxg'(x)

ddxfx+ gx=f'x+g'(x)

Calculation:

By exercise 89:

d2ydx2= d2ydu2dudx2+ dydud2udx2

Differentiatingboth sides

ddxd2ydx2=ddxd2ydu2dudx2+ dydud2udx2

ddxd2ydx2=ddxd2ydu2dudx2+ddxdydud2udx2

By using product rule

d3ydx3=ddxd2ydu2dudx2+d2ydu2ddxdudx2+ddxdydud2udx2+ddxd2udx2dydu

By using chain rule

d3ydx3=ddud2ydu2dudxdudx2+d2ydu2ddxdudx2+ddu

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