Prior Knowledge
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)) $ | $ $ |
| $f(ax+b)$ | $ \frac{1}{a} F(ax+b) + C$ |
Trigonometric Functions
Hello Is this rendering? The function $cos(x)$ is a function you can rewrite as
$$ \sin{(\frac{1}{2}\pi -x)} $$