Definition of the derivative at a point f'(a), with practice problems

Q: What is the difference between f'(a) and f'(x)?

f'(a) is the slope at one point x=a, a number. f'(x) is a function that returns the slope at every x. Plugging x=a into the derivative function gives f'(a).

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

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

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

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 $(a,\,f(a))$ .

Dashed: the secant through two points h apart. As h → 0 the secant approaches the tangent (solid).Drag the dot

Examples

01With $f(x)=7$ , simplify (h≠0): $\displaystyle \frac{f(x+h)-f(x)}{h}$ AnswerHide

Answer $\displaystyle 0$
02Simplify (h≠0): $\displaystyle \frac{(x+h)-x}{h}$ AnswerHide

Answer $\displaystyle 1$
03Find the limit $\displaystyle \lim_{h \to 0}(2x+h)$ AnswerHide

Answer $\displaystyle 2x$
04Which is the definition of the derivative? $\displaystyle f'(x)$ AnswerHide

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

Answer $\displaystyle 2x+h$