# Derivation
The derivative of a function gives us the slope of that function at a specific point.

# Constant Derivatives
Deriving any constant gives us a derivative of $x = 0$

$f(x) = 5$
$f'(x) = 0$

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

$f(x) = x^2$
$f'(x) = x$

$f(x) = x^5$
$f'(x) = 5x^4$

You can also use the power rule to solve $f(x) = \frac{1}{x}$

$$
\begin{flalign}
&f(x) = \frac{1}{x}&\\\
&f(x) = x^{-1}\\
&f('x) = -1x^{-1-1} = -1x^{-2} = \frac{-1x^{-2}}{1}\\
&f'(x) = \frac{-1}{x^2}\\
\end{flalign}
$$

# Constant Multiple Rule
The derivative of $constant * f(x)$  is $constant * f'(x)$. So we can see that the constant doesn't change.
 
$f(x) = 5x^4$
$f'(x) = 5*4x^3 = 20x^3$

$f(x) = 8x^4$
$f'(x) = 32x^3$

$f(x) = 5x^6$
$f'(x) = 30x^5$

# Derive Radical Functions 🤘
$$
\begin{flalign}
&f(x) = \sqrt{x}&\\\
&f(x) = \sqrt[2]{x^1}\\
&f(x) = x^{\frac{1}{2}}\\
&f'(x) = \frac{1}{2}x^{\frac{1}{2}-1} = \frac{1}{2}x^{-\frac{1}{²}}\\
&f'(x) = \frac{1x^{-\frac{1}{2}}}{2}\\
&f'(x) = \frac{1}{2x^{\frac{1}{2}}}\\
&f'(x) = \frac{1}{2\sqrt{x}}
\end{flalign}
$$

# Derive Sine/Cosine

$\frac{d}{dx}[\sin{x}] = \cos{x}$
$\frac{d}{dx}[\cos{x}] = -\sin{x}$

# Product Rule
This rule applies when you try to derive functions that are multiplied.
$\frac{d}{dx}[f*x] = f'*g + f*g'$

# Example
$$
\begin{flalign}
&f(x) = x^3+7x^2-8x+6&\\\
&f'(x) = 3x^2+14x-8
\end{flalign}
$$
$$
\begin{flalign}
&f(x) = 4x^5+3x^4+9x+7&\\\
&f'(x) = 20x^4+12x^3+9
\end{flalign}
$$

$$
\begin{flalign}
&f(x) = 2x^5+5x^3+3x^2+4&\\\
&\text{Find the slope at } x = 2\\
&f'(x) = 10x^4+15x^2+6x\\
&f'(2) = 10(2)^4+15(2)^2+6(2)\\
&f'(2) = 232\\
\end{flalign}
$$

$$
\begin{flalign}
&f(x) = \frac{1}{x^2}&\\\
&f(x) = x^{-2}\\
&f'(x) = -2x^{-3}\\
&f'(x) = \frac{-2}{x^3n }
\end{flalign}
$$

$$
\begin{flalign}
&f(x) = \sqrt[3]{x^5}&\\\
&f(x) = x^{\frac{5}{3}}\\
&f'(x) = \frac{5}{3}x^{\frac{2}{3}}\\
&f'(x) = \frac{5x^{\frac{2}{3}}}{3}\\
&f'(x) = \frac{5\sqrt[3]{x^2}}{3}\\
\end{flalign}
$$

$$
\begin{flalign}
&f(x) = \sqrt[7]{x^4}&\\\
&f(x) = x^{\frac{4}{7}}\\
&f'(x) = \frac{4}{7}x^{-\frac{3}{7}}\\
&f'(x) = \frac{4x^{-\frac{3}{7}}}{7}\\
&f'(x) = \frac{4}{7x^{\frac{3}{7}}}\\
&f'(x) = \frac{4}{7\sqrt[7]{x^3}}
\end{flalign}
$$

$$
\begin{flalign}
&f(x) = (2x-3)^2&\\\
&f(x) = 4x^2-12x+9\\
&f'(x) = 8x - 12  
\end{flalign}
$$

$$
\begin{flalign}
&f(x) = \frac{x^5+6x^4+5x^3}{x^2}&\\\
&f(x) = x^{-2}(x^5+6x^4+5x^3)\\
&f(x) = x^3+6x^2+5x\\
&f'(x) = 3x^2+12x+5
\end{flalign}
$$