# First Derivative: Definition, Test

Share on

First derivative just means taking the derivative (a.k.a. finding the slope of the tangent line) once. It’s usually just shortened to “derivative.”

## First Derivative Test

The first derivative test is one way to study increasing and decreasing properties of functions. The test helps you to:

It’s useful to think of the derivative here as just the slope of the graph. Technically, it’s the slope of the tangent line at a certain point, but simplifying the concept to just increasing or decreasing slopes helps with this particular test. The derivative changes signs (-/+) at points b, c, and d.

Your result from the first derivative test tells you one of three things about a continuous function:

1. If the first derivative (i.e. the slope) changes from positive to negative at a certain point (going from left to right on the number line), then the function has a local maximum at that point. Points b and d on the above graph are examples of a local maximum.
2. If the first derivative changes from negative to positive (going from left to right on the number line), then the function has a local minimum at that point. Point c on the graph is a local minimum.
3. If the first derivative doesn’t change sign at the critical number (going from left to right on the number line), then there is neither a local maximum or a local minimum at that critical number. Point e is one example where the slope does not change sign.

Sample question: Use the first derivative test to find the local maximum and/or minimum for the graph x2 + 6x + 9 on the interval -5 to -1. Graph of x^2 + 6x + 9. The red lines are the slopes of the tangent line (the derivative), which change from negative to positive around x = -3.

Step 1: Find the critical numbers for the function. (Click here if you don’t know how to find critical numbers).

• Taking the derivative: f’= 2x + 6
• Setting the derivative to zero: 0 = 2x + 6
• Using algebra to solve: -6 = 2x then -6/2 = x, giving us x = -3

There is one critical number for this particular function, at x = -3.

Step 2: Choose two values close to the left and right of the critical number. The critical number in this example is x =-3, so we can check x = -2.99 and x = 3.01 (these are arbitrary, but pretty close to -3; you could try -2.999999 and -3.0000001 if you prefer).

Step 3: Insert the values you chose in Step 2, into the derivative formula you found while figuring out the critical numbers in Step 1:
For x = -3.01:
f’ = 2(-3.01) + 6 = -0.02→ a negative slope
For x = -2.99
f’= 2(-2.99) + 6 = 0.02→ a positive slope

Step 4: Compare your answers to the three first derivative test rules (stated in the intro above). The derivative changes from negative to positive around x = -3, so there is a local minimum.

That’s it!

Tip: To check that you found the correct critical numbers, graph your equation. As the graph above clearly shows—you should only find one critical number for this particular equation, at x = -3.

The following table summarizes the application of the first derivative test (f’) and the second derivative test (f”) for drawing graphs.   