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Second derivatives, third derivatives, etc.

This section, in which we differentiate again and again, couldn't be much simpler. We know how to take the derivative of a function already, no problem, easy street. The result, written f'(x), is itself a function of x. But then we can take its derivative again, writing it as f''(x), and calling it the second derivative of f(x). It's like running meat through a grinder a second time before you put the patties on the grill. Makes it so mushy and tender, just the way you like them. (For vegetarians, change the analogy to smoothies in the blender.)

For example, if f(x) = 2x3, then f'(x)= 6x2 and f''(x)=12x.

We'll talk about what second derivatives are good for when we discuss graphing .

Now, as Liz Taylor realized after her second marriage, ``Why stop at two?" We can keep differentiating over and over.

So in this example, f'''(x)=12 and f(4)(x) = 0. Any higher order derivative would also be 0.

In general, f(n)(x) means the nth derivative of f(x).


Example (Trickier)   What is f(101)(x) if $f(x) = \sin x \ ?$

You think we're kidding, right? Take 101 derivatives, we must be out of our minds. But the point is that you don't really have to take that many derivatives. You see, the derivatives cycle through in foursomes.

\begin{eqnarray*}f(x) & = & \sin x \\
f'(x) & = & \cos x \\
f''(x) & = & -\sin x \\
f'''(x) & = & -\cos x \\
f^{(4)} (x) & = & \sin x
\end{eqnarray*}


At the fourth derivative, we are back where we started. Similarly then,

\begin{eqnarray*}f^{(8)}(x) = \sin x, \\
f^{(12)}(x) = \sin x
\end{eqnarray*}


and in fact

\begin{displaymath}f^{(100)}(x) = \sin x
\end{displaymath}

since 4 divides 100. So differentiating one more time, $f^{(101)}(x) = \cos x$.

Awe-inspiring, isn't it?



Next: Velocity and Acceleration: Put Up: Derivatives: How to Find Previous: Derivatives of Trig Functions
Joel Hass
1999-05-26