Basic Derivative Rules

Just as the epsilon-delta definition can be used to prove useful results about limits of simple functions and limits of combinations of functions, the definition of the derivative can be used to prove the following results about derivatives of simple functions and derivatives of combinations of functions. (Note, for each we assume $f'(x)$ and $g'(x)$ exist as needed.)

Together, these rules allow us to take the derivative of a great variety of more complicated functions.

Derivatives of Simple Functions

  1. $\displaystyle{\frac{d}{dx}(k) = 0 \quad \textrm{ for constants } k}$

  1. $\displaystyle{\frac{d}{dx}(x^n) = nx^{n-1}} \quad \textrm{(The Power Rule)}$

  1. $\displaystyle{\frac{d}{dx}(\sin x) = \cos x}$

  2. $\displaystyle{\frac{d}{dx}(\cos x) = -\sin x}$

  3. $\displaystyle{\frac{d}{dx}(\arcsin x) = \frac{1}{\sqrt{1-x^2}}}$

  1. $\displaystyle{\frac{d}{dx}(\sec x) = \sec x \tan x}$

  2. $\displaystyle{\frac{d}{dx}(\csc x) = -\csc x \cot x}$

  3. $\displaystyle{\frac{d}{dx}(\arctan x) = \frac{1}{1+x^2}}$

  1. $\displaystyle{\frac{d}{dx}(\tan x) = \sec^2 x}$

  2. $\displaystyle{\frac{d}{dx}(\cot x) = -\csc^2 x}$

  1. $\displaystyle{\frac{d}{dx}(e^x) = e^x}$

  2. $\displaystyle{\frac{d}{dx}(\ln x) = \frac{1}{x}}$

  1. $\displaystyle{\frac{d}{dx}(a^x) = a^x \ln a}$

  2. $\displaystyle{\frac{d}{dx}(\log_a x) = \frac{1}{x \ln a}}$

Derivatives of Combinations of Functions

The Constant Multiple Rule:
(i.e., constant multipliers can be "pulled out")

 $\displaystyle{\frac{d}{dx} \left[ k f(x) \right] = k \cdot f'(x)}$

The Sum Rule for Derivatives:
(i.e., the derivative of a sum is a sum of the derivatives)

 $\displaystyle{\frac{d}{dx} \left[ f(x) + g(x) \right] = f'(x) + g'(x)}$

The Difference Rule for Derivatives:
(i.e., the derivative of a difference is a difference of the derivatives)

 $\displaystyle{\frac{d}{dx} \left[ f(x) - g(x) \right] = f'(x) - g'(x)}$

The Product Rule for Derivatives:

 $\displaystyle{\frac{d}{dx} \left[ f(x) \cdot g(x) \right] = f'(x)g(x) + f(x)g'(x)}$

The Quotient Rule for Derivatives:

 $\displaystyle{\frac{d}{dx} \left[ \frac{f(x)}{g(x)} \right] = \frac{f'(x)g(x) - f(x)g'(x)}{[g(x)]^2}}$

The Chain Rule for Derivatives:

 $\displaystyle{\frac{d}{dx} \left[ f(g(x)) \right] = f'(g(x)) \cdot g'(x)}$

As a quick example of an application of these rules, suppose we wanted to find the derivative of $\displaystyle{x^3 + 4\sin x}$.

Note,

$$\begin{array}{rclc} \displaystyle{\frac{d}{dx} (x^3 + 4\sin x)} &=& \displaystyle{\frac{d}{dx} (x^3) + \frac{d}{dx} (4\sin x)} & \textrm{(sum rule)}\\ &=& \displaystyle{3x^2 + \frac{d}{dx} (4\sin x)} & \textrm{(power rule)}\\ &=& \displaystyle{3x^2 + 4 \frac{d}{dx} (\sin x)} & \textrm{(constant multiple rule)}\\ &=& 3x^2 + 4 \cos x & \textrm{(derivative of sine)} \end{array}$$