- The derivatives of the basic functions (like $x^2$, $e^x$, $\cos{x}$, etc.)
- The rules of differentiation for functions that have operations (addition, subtraction, multiplication, etc.)
Derivatives of the Basic Functions
The derivatives of the basic functions are given in the following table (here $c \in \mathbb{R}$, $a>0$, $a \neq 1$).
| $f(x)$ | $f'(x)$ |
|---|---|
|
$c$ |
$0$ |
|
$x$ |
$1$ |
|
$x^2$ |
$2x$ |
|
$x^3$ |
$3x^2$ |
|
$\frac{1}{x}$ |
$-\frac{1}{x^2}$ |
|
$\sqrt{x}$ |
$\frac{1}{2\sqrt{x}}$ |
|
$\sqrt[3]{x}$ |
$\frac{1}{3\sqrt[3]{x^2}}$ |
|
$x^n$ |
$nx^{n-1}$ |
|
|
|
|
$\sin{x}$ |
$\cos{x}$ |
|
$\cos{x} $ |
$-\sin{x}$ |
|
$\tan{x}$ |
$\sec^2{x}$ |
|
$\cot{x}$ |
$-\csc^2{x}$ |
|
$\sec{x}$ |
$\sec{x}\tan{x}$ |
|
$\csc{x}$ |
$-\csc{x}\cot{x}$ |
|
|
|
|
$e^{x}$ |
$e^{x}$ |
|
$a^{x}$ |
$a^{x} \cdot \ln{a}$ |
|
$\ln{x}$ |
$\frac{1}{x}$ |
|
$\log_{a}{x}$ |
$\frac{1}{x \cdot \ln{a}}$ |
|
|
|
|
$\sin^{-1}{x}$ |
$\frac{1}{\sqrt{1-x^2}}$ |
|
$\cos^{-1}{x}$ |
$-\frac{1}{\sqrt{1-x^2}}$ |
|
$\tan^{-1}{x}$ |
$\frac{1}{1+x^2}$ |
|
$\cot^{-1}{x}$ |
$-\frac{1}{1+x^2}$ |
|
$\sec^{-1}{x}$ |
$\frac{1}{|x|\sqrt{1-x^2}}$ |
|
$\csc^{-1}{x}$ |
$-\frac{1}{|x|\sqrt{1-x^2}}$ |
Rules of Differentiation
For the functions that involve operations on other functions, there are six rules of differentiation: Coefficient Rule, Addition Rule, Subtraction Rule, Product Rule, Quotient Rule, and Chain Rule.
The Coefficient Rule states that for a constant $k \in \mathbb{R}$ and the function $f(x)$
$ \left( k \cdot f(x) \right)’= k \cdot f'(x) $
Example
$ \left( 3 \cdot \sin{x} \right)’ = 3 \cdot \left( \sin{x} \right)’ = \boxed{3\cos{x}} $
The Addition Rule states that for the functions $f(x)$ and $g(x)$
$ \left( f(x)+g(x) \right)’= f'(x) + g'(x) $
Example
$ \left( x^2 + e^{x} \right)’ = \left( x^2 \right)’ + \left( e^{x} \right)’ = \boxed{ 2x+e^x} $
The Subtraction Rule states that for the functions $f(x)$ and $g(x)$
$ \left( f(x)-g(x) \right)’= f'(x) – g'(x) $
Example
$ \left( x^3 – \ln{x} \right)’ = \left( x^3 \right)’ – \left( \ln{x} \right)’ = \boxed{ 3x^2- \frac{1}{x}} $
The Product Rule states that for the functions $f(x)$ and $g(x)$
$ \left( f(x) \cdot g(x) \right)’= f'(x) \cdot g(x) + g'(x) \cdot f(x) $
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$ \left( x \cdot \tan{x} \right)’ = \left( x \right)’ \cdot \left( \tan{x} \right) + \left( \tan{x} \right)’ \cdot \left( x \right) = \left( 1 \right) \cdot \left( \tan{x} \right) + \left( \sec^2{x} \right) \cdot \left( x \right) = \boxed{ \tan{x} + x \sec^2{x}} $
The Quotient Rule states that for the functions $f(x)$ and $g(x)$
$ \left( \frac{f(x)}{g(x)} \right)’= \frac{f'(x) \cdot g(x) – g'(x) \cdot f(x)}{g^2(x)} $
Example
The Chain Rule states that for the functions $f(x)$ and $g(x)$
$ \left( f(g(x)) \right)’= f'(g(x)) \cdot g'(x) $
Example
$ \sin \left( x^2+3x \right)’= \cos \left( x^2+3x \right) \cdot \left( x^2+3x \right)’ = \boxed{ \left( 2x+3 \right) \cos \left( x^2+3x \right) } $
