Differentiation is the process of finding the gradient / rate of change at a certain point in a graph.

Representing it

The differentiated form of a function is represented as either $f’(x)$, $\frac{\Delta y}{\Delta x}$ or $\frac{dy}{dx}$.

Calculating it

For simple equations

When $f(x) = ax^n$, $f’(x) = nax^(n-1)$.

For trigonometric functions

$sin'(x) = cos(x)$ $cos'(x) = -sin(x)$ $tan'(x) = sec^2(x)$ $cot'(x) = -csc^2(x)$ $sec'(x) = sec(x) \times tan(x)$ $csc'(x) = -csc(x) \times cot(x)$ $arcsin'(x) = \frac{1}{sqrt(1-x^2)}$ $arccos'(x) = \frac{-1}{sqrt(1-x^2)}$ $arctan’(x) = \frac{1}{x^2 + 1}$