Differentiation Rules

Sum Rule:

$\large \frac{d}{d x}(f(x) + g(x)) = \frac{df(x)}{dx} + \frac{d g(x)}{d x}$

Power Rule:

$\large f(x) = ax^b \ \ \ \ \ \ \ \ \ \ \ \ \ f'(x) ax^{b - 1}$

Product Rule:

$\large A(x) = f(x)g(x) \ \ \ \ \ \ \ A'(x) = f(x)g'(x) + g(x)f'(x)$

Chain Rule:

if $h = h(p)$and $p = p(m)$

$\huge \frac{dh}{dm} = \frac{dh}{dp} \cdot \frac{dp}{dm}$

Exponential Functions: Self-Similarity

$\large f(x) = e^x \ \ \ \ \ \ \ \ \ \ \ \ \ \ f'(x) = e^x$

The rate of change of an exponential function is proportional to the function itself.

Trigonometry Functions:

$f(x) \ = \ \ sin(x) \\ f'(x) \ = \ \ cos (x) \\ f''(x) = -sin(x) \\ f^{(3)} \ \ \ = \ -cos(x) \\ f^{(4)} \ \ \ = \ \ \ sin(x)$