# Derivatives

Derivatives is a concept introduced in Calculus where it is the instantaneous rate of change or slope at a given point. The slope of a line can be found by $\frac{rise}{run}$ or $\frac{height}{width}$ or $\frac{\delta y}{\delta x}$.

In the linear equation of $f(x) = 2x$, the slope is 2 because for each incremental change of x the output value would be 2. $\frac{2}{1}$

For curves such as a quadratic, the idea of a slope still applies and it is instead called the derivative. $g(x) = x^2$ has a derivative of 2x because for a change in x such as 2, the resulting output of the function g(2) is twice the input or in this case 4. $g(2) = 4$. $\delta x = 2$ $\delta y = 4$. $\frac{\delta y}{\delta x} = \frac{4}{2}$. We say the instantaneous rate of change or the derivative is 2x. Instead of saying the whole equation’s slope is 2x as we did for f(x) above, we can use the Leibniz notation.
$y=x^2$
$\frac{dy}{dx}=2x$

Further reading: Derivatives Wikipedia