### Home > APCALC > Chapter 3 > Lesson 3.4.2 > Problem3-169

3-169.

Use the definition of a derivative as a limit to write an equation for $f ^\prime$ if $f(x) = 2x + 9$. Use the Power Rule to confirm your answer.

Hana's definition of the derivative: $f'(x) =\lim\limits_{h\rightarrow 0}\frac{f(x+h)-f(x)}{h}$

Substitute $f(x)$ into Hana's definition of the derivative.

Expand the numerator and combine like terms.

Factor out an $h$ from the numerator so you can 'cancel out' the $h$ from the denominator.

Evaluate the limit as $h \rightarrow 0$. This will give you $f ^\prime (x)$.

Confirm that the $f ^\prime (x)$ you found Algebraically agrees with what you get using the Power Rule.