Higher-Order Derivatives

Suppose we want to take the second derivative of $f(x)$. What exactly is that?

The second derivative of $f(x)$ is $f"(x)$. Basically you take the derivative of $f(x)$ and let it be g(x). Then you take the derivative of $g(x)$.

If we wanted the third derivative we could write it as $f'''(x)$.

Notice that the nth derivative of $x^n$ is $n!$.

The reason is that the derivative of $x^n$ is $nx^{n-1}$, the derivative of that is $n(n-1)x^{n-2}$, and you keep going.

What are some applications of higher derivatives?

Now what if we wanted to find the instantaneous acceleration of the object?

We can take the second derivative! In our example, the second derivative would be $10$, so the object's speed is accelerating at $10$ units per second.

If we wanted to find the rate at which the acceleration was growing, we take the third derivative! It turns out to be 0, and that makes sense, because the acceleration is staying constant.

(Video) Higher Order Derivatives in Calculus

by mathtutordvd

Here's a video by mathtutordvd explaining how to find higher order derivatives.

I've attached a worksheet containing some higher-order derivatives. Good luck!