Basic Differentiation Shortcuts (the Simple Power Rule)

The definition of derivative is tedious/difficult to use, particularly as the function being differentiated increases in complexity.

The purpose of this lesson is to begin to develop shortcuts for finding derivatives.

Basic Derivative Shortcuts

Want proofs of the following shortcuts? Read the text!

The Derivative of a Constant is Zero

A constant function graphs as a horizontal line. The slope of a horizontal line is zero.

Examples:

If $\,f(x) = 5\,,$ then $\,f'(x) = 0$
If $\,g(t) = \ln 3\,,$ then $\,g'(t) = 0$
$\displaystyle\frac{d}{dx}(\pi) = 0$
$\displaystyle\frac{d}{dt}(\sqrt{7}) = 0$
$\displaystyle\frac{d}{dw}({\text{e}}^3) = 0$

The Derivative of a Linear Function is the Slope of the Line

Recall that the graph of $\,f(x) = mx + b\,$ is a line with slope $\,m\,.$ Thus, $\,f'(x) = m\,.$

Examples:

If $g(t) = 5t - 1\,,$ then $\,g'(t) = 5$
$\displaystyle\frac{d}{dx} (3 - \frac{2}{7}x) = -\frac 27$

Note that
$$y = 3 - \frac 27x = -\frac 27x + 3$$
graphs as a line with slope $\,-\frac 27\,.$

The Simple Power Rule (How to Differentiate $\,x^n\,$)

SIMPLE POWER RULE how to differentiate $\,x^n$

For all real numbers $\,n\,$: $$\frac d{dx} x^n = nx^{n-1}$$

For the More Advanced Reader

Here's a more precise statement of the Simple Power Rule:

If $\,n\,$ is a real number, and $\,I\,$ is any interval on which both $\,x^n\,$ and $\,nx^{n-1}\,$ are defined, then $\,x^n\,$ is differentiable on the interval $\,I\,,$ and:

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

Power Functions

Power functions are functions that can be written in the form $\,x^n\,,$ for some real number $\,n\,.$

Examples of power functions:

$x^2$
$x^3$
$x^{1.4}$
$x^\pi$
$\displaystyle\frac 1x = x^{-1}$
$\displaystyle\frac 1{x^2} = x^{-2}$
$\sqrt x = x^{1/2}$
$\root 3\of {x^2} = x^{2/3}$

The Simple Power Rule tells how to differentiate power functions.

To differentiate $\,x^n\,$:

Bring the exponent down front
Decrease the original exponent by $\,1$

Examples

Example 1

If $f(x) = x^3\,,$ then $f'(x) = 3x^2\,.$

Example 2

If a function is given in radical form, then it's conventional to write the derivative in radical form.

$$ \begin{align} &\frac{d}{dx} \sqrt x\cr\cr &\quad = \frac d{dx} x^{1/2}\cr &\qquad \text{(rename as a power function)}\cr\cr &\quad = \frac{1}{2} x^{-1/2}\cr &\qquad \text{(use the Simple Power Rule)}\cr\cr &\quad = \frac{1}{2x^{1/2}}\cr &\qquad \text{(rename)}\cr\cr &\quad = \frac{1}{2\sqrt x}\cr &\qquad \text{(write in radical form)} \end{align} $$

Example 3

Be sure that after you've used the $\,\frac{d}{dt}\,$ operator (that is, after you've taken the derivative), that the $\,\frac{d}{dt}\,$ operator is gone!

$$ \begin{align} &\frac{d}{dt} \frac{1}{\sqrt t}\cr\cr &\quad = \frac d{dt} t^{-1/2}\cr &\qquad \text{(rename function)}\cr\cr &\quad = -\frac{1}{2} t^{-3/2}\cr &\qquad \substack{\text{(use Simple Power Rule;}\\ \text{$\frac{d}{dt}$ is now gone)}}\cr\cr &\quad = -\frac{1}{2t^{3/2}}\cr &\qquad \text{(rename derivative)}\cr\cr &\quad = \frac{-1}{2\sqrt {t^3}}\cr &\qquad \text{(write in radical form)} \end{align} $$

Example 4

Always try to write the derivative in a form that matches—as closely as possible—the original function.

$$ \begin{align} \frac{d}{dt} t\root 3\of t &= \frac{d}{dt} t^1 t^{1/3}\cr\cr &= \frac d{dt} t^{4/3}\cr\cr &= \frac 43 t^{1/3}\cr\cr &= \frac 43\root 3\of t\cr\cr &= \frac {4\root 3\of t}{3} \end{align} $$

(Either of these last two names for the derivative is good!)