Differentiation Rules

Table of contents

1. Differentiation Rules
   1. Basic Rules
   2. Chain Rule
   3. References

Objectives

The Basic Rules

• List of basic rules
• Examples

Chain Rule

• Definition
• Applications

Higher deriviation

• Definition

Basic Rules

Constant law

$\left(cf(x)\right)^\prime=cf^\prime\left(x\right)$

$\left(\lim_{x\to a}{cf(x)} = c\lim_{x\to a}{f(x)} \right)$

Sum/Diff law

$\left(f(x) \pm g(x)\right)^\prime=f^\prime\left(x\right)\pm g^\prime\left(x\right)$

$\left(\lim_{x\to a}{\left[f(x)\pm g(x)\right]} = \lim_{x\to a}{f(x)}\pm\lim_{x\to a}{g(x)}\right)$

Power law

$\left(x^n\right)^\prime = nx^{n-1}$

$\left(\lim_{x\to a}{\left[f(x)\right]^n} = \left[\lim_{x\to a}{f(x)}\right]^n\right)$

Product law

$\left(fg\right)^\prime=f^\prime g + g^\prime f$

$\left(\lim_{x\to a}{\left[f(x)\cdot g(x)\right]} = \lim_{x\to a}{f(x)}\cdot\lim_{x\to a}{g(x)}\right)$

Quotient law

$\left(\frac{f}{g}\right) = \frac{f^\prime g - fg^\prime}{g^2}$

$\left(\lim_{x\to a}{\left[\frac{f(x)}{g(x)}\right]} = \frac{\lim_{x\to a}{f(x)}}{\lim_{x\to a}{g(x)}}\right)$

Chain Rule

Suppose there are two functions $f$ and $g$, both are differentiable:

1. If there is a function $h: h(x) = g(f(x))$, then $h$ is called a composite function
2. The composite function $h$ is denoted as: $h=g\circ f \quad \text{or} \quad (g\circ f)(x) = g(f(x))$
3. The derivative of the composite function: $h^\prime(x) = g^\prime(f(x))f^\prime(x) \quad\text{or}\quad \frac{dh}{dx} = \frac{dh}{df}*\frac{df}{dx}$

References