Chapter 10.3, Problem 97E

### Calculus (MindTap Course List)

11th Edition
Ron Larson + 1 other
ISBN: 9781337275347

Chapter
Section

### Calculus (MindTap Course List)

11th Edition
Ron Larson + 1 other
ISBN: 9781337275347
Textbook Problem

# True or False? In Exercises 97-100, determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.If x = f ( x )   and   y = g ( t ) ,   then   d 2 y d x 2 = g " ( t ) f " ( t )

To determine
Whether the statement “if x=f(t) and y=g(t), then d2ydx2=g(t)f(t)” is true or false.

Explanation

Consider the parametric equations, x=f(t) and y=g(t).

Differentiate x=f(t) with respect to t. To get,

dxdt=fâ€²(t)

Differentiate y=g(t) with respect to t. To get,

dydt=gâ€²(t)

To find out the ratio of the both of differentiation. So,

dydx=dydtdxdt=gâ€²(t)fâ€²(t)

Again, differentiate the above equation again with respect to x.

d2ydx2=ddt(dydx)dxdt=ddt(gâ€²(t)fâ€²(

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