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## Basic Derivatives

· Power rule for derivatives

The power rule for differentiation states that if $n$ is a real number and $f(x) = x^n$, then $f'(x) = nx^{n-1}$

$\frac{d}{dx}\left(x^a\right)=ax^{\left(a-1\right)}$
· Sum Rule

The derivative of a sum of two functions is the sum of the derivatives of each function

$\frac{d}{dx}\left(a+b\right)=\frac{d}{dx}\left(a\right)+\frac{d}{dx}\left(b\right)$
· Derivative of a Constant

The derivative of the constant function ($[c]$) is equal to zero

$\frac{d}{dx}\left(c\right)=0$

The derivative of a function multiplied by a constant ($[c]$) is equal to the constant times the derivative of the function

$\frac{d}{dx}\left(cx\right)=c\frac{d}{dx}\left(x\right)$
· Derivative of the linear function

The derivative of the linear function is equal to $1$

$\frac{d}{dx}\left(x\right)=1$

## Derivatives of trigonometric functions

· Derivative of the cosine function

The derivative of the cosine of a function is equal to minus the sine of the function times the derivative of the function, in other words, if $f(x) = \cos(x)$, then $f'(x) = -\sin(x)\cdot D_x(x)$

$\frac{d}{dx}\left(\cos\left(x\right)\right)=-\sin\left(x\right)$
· Derivative of the sine function

The derivative of the sine of a function is equal to the cosine of that function times the derivative of that function, in other words, if ${f(x) = \sin(x)}$, then ${f'(x) = \cos(x)\cdot D_x(x)}$

$\frac{d}{dx}\left(\sin\left(x\right)\right)=\cos\left(x\right)$
$\lim_{x\to\:0}\left(\frac{1-\cos\left(x\right)}{x^2}\right)$

Limits

### Time to solve it:

~ 0.04 s (SnapXam)