The derivative function f'(x) from the definition, with practice problems

f'(x)=\lim_{h\to 0}\frac{f(x+h)-f(x)}{h}

The derivative at a point was the slope at one $x$ . If you keep $a$ as $x$ and take the limit, you get a function that returns the slope at every $x$ : the derivative function $f'(x)$ .

f'(x)=\lim_{h\to 0}\frac{f(x+h)-f(x)}{h}

Every formula that follows is a shortcut so you do not have to redo this definition each time.

The derivative f'(x)=2x records the slope of f(x)=x² at each point: negative on the left, positive on the right.Drag the dot

Examples

01What is f'(x)? $\displaystyle f(x)=x^2+2$ AnswerHide

Answer $\displaystyle 2x$
02What is f'(6)? $\displaystyle f'(x)=2x$ AnswerHide

Answer $\displaystyle 12$
03What goes in □? $\displaystyle \frac{2xh+h^2}{h} = 2x+\square$ AnswerHide

Answer $\displaystyle h$
04What is f'(x)? $\displaystyle f(x)=x^2$ AnswerHide

Answer $\displaystyle 2x$
05What is f'(x)? $\displaystyle f(x)=x^2+5$ AnswerHide

Answer $\displaystyle 2x$
06What is f'(x)? $\displaystyle f(x)=x^2+6$ AnswerHide

Answer $\displaystyle 2x$

FAQ

Q1Do I have to use the definition every time?Show answerHide

No. The results of doing it once are the rules (the power rule and so on), and you normally use those. You go back to the definition for functions the rules do not cover, or in proofs.

Practice this pattern in the app, 5 problems a day.

Before this

The derivative at a point, f'(a)

This topic

The derivative function f'(x)

x^n from the definition Sign of f' and increasing/decreasing Product rule Chain rule

Published 2026-09-03 · Updated 2026-09-05