# Interpreting the Sign of the First Derivative (Increasing and Decreasing Functions)

## Key Questions

• #### Answer:

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!

• #### Answer:

If ${x}_{0} < {x}_{1}$ then $f \left({x}_{0}\right) < f \left({x}_{1}\right)$

#### Explanation:

The meaning is that you have a function with positive slope in every point of Dom.

Starting from a ${x}_{0}$ and move to right, the graph of function is moving up at the same time