The derivative function f'(x)

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 $f'(x)$.

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

Examples

1. 01What is f'(x)?$\displaystyle f(x)=x^2+2$AnswerHide

   Answer$\displaystyle 2x$

   Steps

   1. $\displaystyle f(x)$$\displaystyle = x^2 +2$
   2. Differentiate
   3. $\displaystyle f'(x)$$\displaystyle = 2x + 0 = 2x$
2. 02What is f'(6)?$\displaystyle f'(x)=2x$AnswerHide

   Answer$\displaystyle 12$

   Steps

   1. $\displaystyle f(x)$$\displaystyle = x^2$
   2. Differentiate
   3. $\displaystyle f'(x)$$\displaystyle = 2x$
   4. Put x = 6 in
   5. $\displaystyle f'(6)$$\displaystyle = 2 \cdot (6) = 12$
3. 03What goes in □?$\displaystyle \frac{2xh+h^2}{h} = 2x+\square$AnswerHide

   Answer$\displaystyle h$

   Steps

   1. $\displaystyle \frac{2xh+h^2}{h}$$\displaystyle = \frac{h(2x+h)}{h} = 2x+h$
   2. Let h approach 0
   3. $\displaystyle (h \to 0)$$\displaystyle \Rightarrow f'(x) = 2x$
4. 04What is f'(x)?$\displaystyle f(x)=x^2$AnswerHide

   Answer$\displaystyle 2x$

   Steps

   1. $\displaystyle f(x)$$\displaystyle = x^2$
   2. Differentiate
   3. $\displaystyle f'(x)$$\displaystyle = 2 x^{2-1} = 2x$
5. 05What is f'(x)?$\displaystyle f(x)=x^2+5$AnswerHide

   Answer$\displaystyle 2x$

   Steps

   1. $\displaystyle f(x)$$\displaystyle = x^2 +5$
   2. Differentiate
   3. $\displaystyle f'(x)$$\displaystyle = 2x + 0 = 2x$
6. 06What is f'(x)?$\displaystyle f(x)=x^2+6$AnswerHide

   Answer$\displaystyle 2x$

   Steps

   1. $\displaystyle f(x)$$\displaystyle = x^2 +6$
   2. Differentiate
   3. $\displaystyle f'(x)$$\displaystyle = 2x + 0 = 2x$

FAQ