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

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

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

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

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

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

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

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