The first derivative of a function is really important because it helps us understand where that function is going up or down. Here’s a simpler look at it:

1. What is the First Derivative?
   The first derivative, which we write as $f'(x)$, shows us the slope of the function $f(x)$ at any point.

   • If $f'(x) > 0$, it means the function is going up.
   • If $f'(x) < 0$, it means the function is going down.

2. Finding Where the Function is Increasing:
   To see where the function is climbing, we look for parts where the first derivative is positive.

   • For example, if $f'(x) = 3x - 2$, we want to find when this is greater than zero:
     $3x - 2 > 0$
     This simplifies to:
     $x > \frac{2}{3}$
     So, the function $f(x)$ is going up when $x$ is greater than $\frac{2}{3}$.

3. Finding Where the Function is Decreasing:
   To know where the function is falling, we check where the first derivative is negative.

   • Using the same example, we want to find when $3x - 2$ is less than zero:
     $3x - 2 < 0$
     This leads us to:
     $x < \frac{2}{3}$
     This means that $f(x)$ is going down when $x$ is less than $\frac{2}{3}$.

In short, looking at the first derivative is like having a map for the function. It shows us where we can expect the graph to go up or down. This is really helpful for drawing graphs or understanding how the function behaves!