# Constants, Identities, and Variations

## Constants, Identities, and Variations

(Some Essential Constants in Identities and Variations using $\large\rm\LaTeX$)

#### Calculus:

If we have the following conditions:

1. $f(x)$ is continuous on $[a,b]$,
2. $f(a)$ and $f(b)$ are of different signs,
Then there exists a point $\xi\in(a,b)$ such that $f(\xi)=0$.

Power Rule:
$\frac{d(x^n)}{dx}=nx^{n-1}$

Sum Rule:
$\frac{d}{dx}\bigl(g(x)+h(x)\bigr)=\frac{dg}{dx}+\frac{dh}{dx}$

Product Rule:
$\frac{d}{dx}\bigl(f(x)g(x)\bigr)=f(x)g^\prime(x)+g(x)f^\prime(x)$

Chain Rule:
$\frac{d}{dx}\bigl(g(h(x)\bigr)=\frac{dg}{dh}\bigl(h(x)\bigr)+\frac{dh}{dx}(x)$