Evaluating a derivative is the most basic application of your derivative. By evaluating a derivative, you are using your slope equation to actually find the slope at a single point. You are finding that specific slope of the tangent line, that specific instantaneous rate of change. It really just means plugging values into your derivative. Sometimes you will just be asked to evaluate a derivative at a specific value, and other times evaluating your derivative will just be a single step in a larger process.
This is one of the topics on the AP Calculus exam that you can use the calculator on at times. I will provide you with two methods on how to solve these problems. First, I will show you how to do these by hand, and then I will show you how to accomplish the same task on your calculator.
Meaning: Derivative Value
When you evaluate the derivative, ${\textcolor[rgb]{}{f}}^{\textcolor[rgb]{}{\prime}}\textcolor[rgb]{}{\left(}\textcolor[rgb]{}{x}\textcolor[rgb]{}{\right)}$, at a specific x-value , you are finding out several pieces of information.
1) You have found the slope of the tangent line to your original equation, $\textcolor[rgb]{}{f}\textcolor[rgb]{}{\left(}\textcolor[rgb]{}{x}\textcolor[rgb]{}{\right)}$, at that single x-value . Remember the tangentlineis a line that just grazes or touches our original graph at that single point, x-value , we are evaluating at. The tangentline is separate from the original equation, but they share that one tangent point, $\left(\textcolor[rgb]{}{x}\textcolor[rgb]{}{,}\textcolor[rgb]{}{y}\right)$, where they intersect.
2) You have found the instantaneous rate of change of your original equation, $\textcolor[rgb]{}{f}\textcolor[rgb]{}{\left(}\textcolor[rgb]{}{x}\textcolor[rgb]{}{\right)}$, at that single x-value . At that single instance in time, we now know how the original graph is behaving. Another way of saying it would be we have found the slope of our original equation, $\textcolor[rgb]{}{f}\textcolor[rgb]{}{\left(}\textcolor[rgb]{}{x}\textcolor[rgb]{}{\right)}$, at that x-value in time.
3) You can then use the value of the instantaneous rate of change to determine if your original equation, $\textcolor[rgb]{}{f}\textcolor[rgb]{}{\left(}\textcolor[rgb]{}{x}\textcolor[rgb]{}{\right)}$, is increasing, decreasing, or not changing at that single x-value .
The best way to keep track of all of this is by using this derivative triangle tool.
Note :
– When working with derivatives keep in mind that everything boils down to plus, minus, and zero; positive, negative, zero.
– The specific value of the derivative has meaning, the value represents the steepness of the slope.