# How do you know a function is increasing?

Mar 9, 2018

It will be increasing when the first derivative is positive.

#### Explanation:

Take the example of the function $f \left(x\right) = {e}^{{x}^{2} - 1}$.

The first derivative is given by $f ' \left(x\right) = 2 x {e}^{{x}^{2} - 1}$ (chain rule). We see that the derivative will go from increasing to decreasing or vice versa when $f ' \left(x\right) = 0$, or when $x = 0$.

Whenever you have a positive value of $x$, the derivative will be positive, therefore the function will be increasing on $\left\{x | x > 0 , x \in \mathbb{R}\right\}$.

The graph confirms

Hopefully this helps!