**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:

- Find the intervals where a function is decreasing or increasing.
- Identify local minima and local maxima.
- Sketch a graph without the aid of a graphing calculator (although you can also use “rise over run” to sketch the graph of a derivative).

**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.

Your result from the first derivative test tells you one of

**three things**about a continuous function:

- 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. - 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. - 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 x ^{2} + 6x + 9 on the interval -5 to -1.*

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.**

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