Question

How do you use the definition of a derivative to find the derivative of `f(x)=-4x-9`?

Answer 1

Apply the Difference Quotient as follows:
`(-4(x+h)-9-(-4x-9))/h`

Explanation:

Continue to simplify:
`(-4(x+h)-9-(-4x-9))/h`=`(-4x-4h-9+4x+9)/h`

Combine like terms to get: `(-4h)/h = -4`.

The derivative of this function is the constant, -4, since the slope of the linear function is 4.

For polynomial functions of higher degree (and many other functions), your last step is to take the limit of the expression as h approaches 0. Your last step did not have a term with h, so the limit of that constant is itself.