Derivatives

The formal definition of a derivative is

$\underset{h\to 0}{\text{lim}}\frac{f\left(x+h\right)-f\left(x\right)}{h}$

Stated simply, the derivative is an equation of a function that represents the slope of another line.

Here is an example of using the formal definition of the derivative to calculate the derivative of x to the second power.

1. $f\left(x\right)={x}^{2}$

$f\prime \left(x\right)=\underset{h\to 0}{\text{lim}}\frac{f\left(x+h\right)-f\left(x\right)}{h}$
3. Put f(x) into the formal definition.

$f\prime \left(x\right)=\underset{h\to 0}{\text{lim}}\frac{{\left(x+h\right)}^{2}-{x}^{2}}{h}$
4. Simplify.

$f\prime \left(x\right)=\underset{h\to 0}{\text{lim}}\frac{{x}^{2}+2xh+{h}^{2}-{x}^{2}}{h}$
5. Subtract x^2.

$f\prime \left(x\right)=\underset{h\to 0}{\text{lim}}\frac{2xh+{h}^{2}}{h}$
6. Cancel out h.

$f\prime \left(x\right)=\underset{h\to 0}{\text{lim}}2x+h$
7. Evaluate the limit.

$f\prime \left(x\right)=2x$

The final answer is 2x as you can see. Fortunately, the derivative can be found using a shortcut called the Power Rule.

Power Rule

The Power Rule is a shortcut for calculating the derivative of a term. Note: The notation d/dx will be used to signify "the derivative of". Here's the equation.

$\frac{d}{\mathrm{dx}}{x}^{n}=n{x}^{n-1}$

Therefore, the derivative of 3x^2 would be

$\frac{d}{\mathrm{dx}}3{x}^{2}=6x$

Derivative Table

$\begin{array}{ccc}\text{Antiderivative}& \text{Function}& \text{Derivative}\\ -\mathrm{cos}x& \mathrm{sin}x& \mathrm{cos}x\\ \mathrm{sin}x& \mathrm{cos}x& -\mathrm{sin}x\\ -\mathrm{ln}\left(\mathrm{cos}x\right)& \mathrm{tan}x& {\mathrm{sec}}^{2}x\\ -\mathrm{ln}|\mathrm{csc}x+\mathrm{cot}x|& \mathrm{csc}x& -\left(\mathrm{csc}x\right)\left(\mathrm{cot}x\right)\\ \mathrm{ln}|\mathrm{sec}x+\mathrm{tan}x|& \mathrm{sec}x& \left(\mathrm{sec}x\right)\left(\mathrm{tan}x\right)\\ \mathrm{ln}|\mathrm{sin}x|& \mathrm{cot}x& -{\mathrm{csc}}^{2}x\end{array}$