   Chapter 2, Problem 23RQ

Chapter
Section
Textbook Problem

Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. d 2 y d x 2 = ( d y d x ) 2

To determine

Whether the statement, “d2ydx2=(dydx)2” is true or false.

Explanation

Given that (dydx)2 is a first derivative squared and d2ydx2 is a second derivative.

The given statement is false since the below example disproves the given statement.

Let y=f(x) and f(x)=2x.

Formula used:

The derivative of a function f , denoted by f(x) is,

f(x)=limh0f(x+h)f(x)h (1)

Calculation:

Obtain the first derivative of f(x)=2x.

Compute f(x) by using the equation (1).

f(x)=limh0f(x+h)f(x)h=limh0(2(x+h))(2x)h=limh02x+2h2xh=limh02hh

Since the limit h approaches zero but not equal to zero, cancel the common term h from both the numerator and the denominator,

f(x)=limh02=2

Thus, the derivative of the function f(x) is 2. That is, dydx=2

Take the square of the derivative, (dydx)2=4 (1)

Obtain the derivative of f(x)=2

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