BuyFind

Single Variable Calculus: Concepts...

4th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781337687805
BuyFind

Single Variable Calculus: Concepts...

4th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781337687805

Solutions

Chapter 3.4, Problem 93E
To determine

Explain the proof of the given relation using given data.

Expert Solution

Explanation of Solution

Given:

The given relation is d2ydx2=d2ydu2(dudx)2+dydud2udx2 , where y=f(u) and u=g(x) .

Calculation:

  y=f(u) and u=g(x)

Apply chain rule.

  df(a)dx=dfdadadx

  dydx=dydududxd2ydx2=ddx(dydududx)

Apply product rule (fg)'=gf'+fg' .

  d2ydx2=dudxddx(dydu)+dyduddx(dudx)d2ydx2=d2ydu2(dudx)2+dydud2udx2=R.H.S

Hence d2ydx2=d2ydu2(dudx)2+dydud2udx2 proved.

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