Question

Find the derivative of the function using the definition of the derivative.

f(x) = (1/x² ) - 64

A) f'(x) = -2/x³
B) f'(x) = -2/x³
C) f'(x) = 2/x³
D) f'(x) = -1/x³

Answer 1

To find the derivative of the function using the definition of the derivative, apply the limit definition and simplify the expression.

Step-by-step explanation:

To find the derivative of the function using the definition of the derivative, we need to use the limit definition. The derivative of a function f(x) at a point x is defined as the limit of the difference quotient as h approaches 0:

f'(x) = lim(h→0) [f(x+h) - f(x)]/h

In this case, the function is f(x) = (1/x^2) - 64. Let's apply the definition:

1. Start by substituting f(x+h) and f(x) into the difference quotient.
2. Simplify the expression by combining like terms.
3. Take the limit as h approaches 0 to find the derivative.

After following these steps, we find that the derivative of the function f(x) = (1/x^2) - 64 is f'(x) = -2/x^3.