The derivative at a point, f'(a)

The derivative at a point, $f'(a)$, is where the average rate of change ends up as you shrink the interval around $x=a$.

The numerator is $f(a+h)-f(a)$, the denominator is $h$. Cancel $h$ first, then let $h\to 0$, and you get the slope of the tangent line at $(a,\,f(a))$.

Examples

1. 01Find the limit$\displaystyle \lim_{h \to 0}(2x+h)$AnswerHide

   Answer$\displaystyle 2x$

   Steps

   1. Just let $h$ shrink to 0; the $h$ term disappears
   2. $\displaystyle \lim_{h \to 0}(2x+h)$$\displaystyle = 2x$
2. 02With $f(x)=7$, simplify (h≠0):$\displaystyle \frac{f(x+h)-f(x)}{h}$AnswerHide

   Answer$\displaystyle 0$
3. 03Simplify (h≠0):$\displaystyle \frac{(x+h)-x}{h}$AnswerHide

   Answer$\displaystyle 1$
4. 04Which is the definition of the derivative?$\displaystyle f'(x)$A$\displaystyle \lim_{h \to 0}\frac{f(x+h)+f(x)}{h}$B$\displaystyle \lim_{h \to 0}\frac{f(x+h)-f(x)}{h}$C$\displaystyle \frac{f(x+h)-f(x)}{h}$D$\displaystyle \lim_{h \to 0}\frac{f(x)-f(x+h)}{h}$AnswerHide

   Answer$\displaystyle \lim_{h \to 0}\frac{f(x+h)-f(x)}{h}$
5. 05Simplify (h≠0):$\displaystyle \frac{(x+h)^2-x^2}{h}$AnswerHide

   Answer$\displaystyle 2x+h$

FAQ