The function's first derivative tells us many things about the math function itself. In this post, we'll explore its most essential meaning.

## The First Derivative is the Slope

Let's start by defining what the first derivative is. At its basic, it tells us the steepness of the function \(f\). In other words, it is the slope \(m\) of the \(f\).

\(y'=\frac{dy}{dx}=f^{\prime}(x)=m\)

To illustrate, consider the quadratic function \(f(x)=y=5x^2+3\). The first derivative is equal to \(\frac{dy}{dx}=10x\). It means that the function's slope is equal to \(10x\). With this expression, we can determine the slope of the tangent line at any point along the curve. Say we want to find the slope at \(x=1\), we substitute it to \(\frac{dy}{dx}=10x\)

- \(\frac{dy}{dx}=10x\)
- \(\frac{dy}{dx}=10(1)=10\)

At \(x=1\), the slope of the parabola at that point is equal to 10. We can further verify this by drawing a tangent line at that point and solving for the tangent line's slope using a graphing utility.

## Interpreting First Derivative

Aside from telling us the slope, we can interpret if the given function increases, decreases, or remains the same with the first derivative.

To illustrate these three conditions, consider a math function \(f\) and two values \(x_1\) and \(x_2\) so that \(x_2 \gt x_1\). We'll also explore the slopes of the function along these points. We can summarize increasing, decreasing, and constant functions using the table below:

- If \(f(x_1) \gt f(x_2)\), then the graph is moving downward or decreasing. The slope of the tangent lines for decreasing functions is negative.
- If \(f(x_1) \lt f(x_2)\), then it is moving upward or increasing. The slope of the tangent lines for increasing functions is positive.
- If \(f(x_1) = f(x_2)\), then the graph is constant. The slope is zero for constant functions.

With this table, we can relate increasing, decreasing, and constant functions using the sign convention of \(\frac{dy}{dx}\):

- If \(\frac{dy}{dx}\) is negative on a certain interval, then the function is decreasing.
- If \(\frac{dy}{dx}\) is positive on a certain interval, then it is increasing.
- If \(\frac{dy}{dx}\) equals zero on a certain interval, then it is zero.

## Summary

The first derivative tells us the steepness of the function \(f\). In other words, it is the slope \(m\) of the \(f\).

Aside from telling us the slope, we can interpret if the given function increases or decreases with the first derivative.

If \(\frac{dy}{dx}\) is negative on a certain interval, then the function is decreasing.

If \(\frac{dy}{dx}\) is positive on a certain interval, then it is increasing.

If \(\frac{dy}{dx}\) equals zero on a certain interval, then it is zero.