Applications of the Definitions of the Derivative

Check out my new book on sequences and series!

You can read it on your Kindle, iPad, or Android device.

Click to see Calculus Sequences and Series: Problems and Solutions

Now we’ve seen two different definitions of the derivative (one here and another here), let’s look at one important application.

Sometimes teachers give you a tricky limit that’s really a derivative in disguise. This question is from the sample questions for the AP Calculus exam:

“What is

$\displaystyle \lim_{h\rightarrow 0}\frac{\cos\left(\frac{3\pi}{2}+h\right)-\cos\left(\frac{3\pi}{2}\right)}{h}$ ?”

Then there are some multiple choice answers. This one looks tough, because if you just plug in $h=0$ the denominator and numerator are both zero. What can we do?

The trick is to recognize that this limit is just the definition of the derivative of cosine evaluated at $\frac{3\pi}{2}$ . Do you see it? If you have trouble, go back to the definition, use $f(x)=\cos x$ and then plug in $\frac{3\pi}{2}$ for $x$ .