Derivatives as a function

Question

Find the derivative of f(x)= 8/x² using the limit definition of the derivative.

Answer 1

Find the derivative of f(x)= 8/x² using the limit definition of the derivative.

f'(x) = lim_h→0[f(x+h) – f(x)]/h is the limit definition of the derivative, also called the difference quotient.

For f(x)= 8/x² this limit is

f'(x) = lim_h→0[8/(x+h)² – 8/x²]/h

First let's simplify the numerator:

8/(x+h)² – 8/x² = [8x² – 8(x+h)²]/x²(x+h)²

Since

8x² – 8(x+h)² = 8x² – 8x² – 8(2hx + h²) = – 8(2hx + h²)

This leads to

8/(x+h)² – 8/x² = –8(2hx + h²)/x²(x+h)²

Now divide this simplified numerator by h to get a simplified version of the difference quotient:

f'(x) = lim_h→0[–8(2hx + h²)/hx²(x+h)²]

The numerator and denominator have h as a common factor, cancellation givies

f'(x) = lim_h→0[–8(2hx + h²)/hx²(x+h)²]

= lim_h→0[–8(2x + h)/x²(x+h)²]

Now lim_h→0[–8(2x + h)/x²(x+h)²] is just –8(2x + h)/x²(x+h)² evaluated at h = 0

–8(2x + 0)/x²(x+0)² = –8(2x)/x²(x)² = –16x/x⁴ = –16/x³

Answer

The derivative of f(x)= 8/x² evaluated using the limit definition of the derivative is

f'(x) = –16/x³

Answer 2

The limit definition of the derivative is:

We are told that f(x) is "8/x²"

Then f(x + h) = 8/(x + h)²

To simplify f(x + h) - f(x) you need to get a common denominator: