Prior Knowledge

July 21, 2025

Differentiation and Integration

There are a few standard differentiation and integration formulas that you should know. These are the most important ones. I’m using the convention that:

$C$ is the constant of integration.
$n$ is a constant that is not equal to $- 1$ .
$a$ is a constant that is greater than $0$ .
$x$ is the variable of differentiation or integration.

Differentiation	Formula	Integration
$n x^{n - 1}$	$x^{n}$	$\frac{1}{n + 1} x^{n + 1} + C$
$\ln (a) a^{x}$	$a^{x}$	$\frac{1}{\ln (a)} a^{x} + C$
$\cos (x)$	$\sin (x)$	$- \cos (x) + C$
$\frac{1}{x}$	$\ln (x)$	$x \ln (x) - x + C$

There are also some rules on how you can combine these formulas. For now, I will use the convention that:

$f^{'} (x)$ is the derivative of $f (x)$ .
$F (x)$ is the antiderivative of $f (x)$ .

Differentiation	Formula	Integration
$f^{'} (x) + g^{'} (x)$	$f (x) + g (x)$	$F (x) + G (x) + C$
$n \cdot f^{'} (x)$	$n \cdot f (x)$	$n \cdot F (x) + C$
$f^{'} (x) \cdot g (x) + f (x) \cdot g^{'} (x)$	$f (x) \cdot g (x)$
$\frac{f^{'} (x) \cdot g (x) + f (x) \cdot g^{'} (x)}{(f (x))^{2}}$	$\frac{f (x)}{g (x)}$
$f^{'} (g (x)) \cdot g^{'} (x)$	$f (g (x))$
	$f (a x + b)$	$\frac{1}{a} F (a x + b) + C$

Trigonometric Functions

Hello Is this rendering? The function $c o s (x)$ is a function you can rewrite as

$\sin (\frac{1}{2} π - x)$