Chapter 2.5, Problem 89E

### Single Variable Calculus

8th Edition
James Stewart
ISBN: 9781305266636

Chapter
Section

### Single Variable Calculus

8th Edition
James Stewart
ISBN: 9781305266636
Textbook Problem

# If y = f(u) and u = g(x), where f and g are twice differentiable functions, show that d 2 y d x 2 = d 2 y d u 2 ( d u d x ) 2 + d y d u d 2 u d x 2

To determine

To show: The derivative d2ydx2=d2ydu2(dudx)2+dydud2udx2.

Explanation

Result used: Chain Rule

If y=f(u) and u=g(x) are both differentiable function, then dydx=dydududx (1)

Derivative Rule:

Product Rule: ddx[f(x)g(x)]=f(x)ddx[g(x)]+g(x)ddx[f(x)]

Proof:

Let y=f(u) and u=g(x),Â  where f and g are twice differentiable functions.

d2ydx2=ddx(dydx)=ddx(dydududx)Â Â Â Â Â Â Â Â (ChainÂ Rule(1))

Apply the product rule (1),

d2ydx2=dyduddx

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