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.
$\frac{d}{dx}[f*x] = f'g + fg'$

Example

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

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

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

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

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

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

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