# Second Derivatives and Beyond

### Topics

We're learning calculus, so, by definition, we're in the prime of our lives. Things are great. Things are awesome.

We can take third derivatives, fourth derivatives, and so on. Writing *f *" for the second derivative of *f* is fine, and writing *f *"' for the third derivative isn't so bad. Who wants to spend the prime of their life writing...primes? Few math people write *f *""' for the fifth derivative. It's too much to keep track of.

Instead, we keep track of which derivative we're on by writing what looks like an exponent but has little parentheses around it. The first derivative of the function *f *(*x*) can be written

*f *^{(1)}(*x*)

For the second derivative of *f *(*x*) we write

*f *^{(2)}(*x*) ,

and for the fifth derivative of *f *(*x*) we write

*f *^{(5)}(*x*) .

**Be Careful:** f ^{2}(*x*) means the square of the function *f *(*x*), while *f *^{(2)}(*x*) means the second derivative of *f *(*x*). Those parentheses are important!

The derivative of a polynomial is also a polynomial. Each time we take a derivative, the degree of the polynomial goes down by 1.

Example. Let *f *(*x*) = x^{6}. We take a derivative, and the degree goes down by 1:

*f *'(*x*) = 6*x*^{5}.

We take another derivative, and the degree goes down by 1 again:

*f *"(*x*) = 30*x*^{4}.

We take another, and the degree goes down again:

*f *^{(3)}(*x*) = 120*x*^{3}.

And so on:

*f *^{(4)}(*x*) = 360*x*^{2}

and so on:

*f *^{(5)}(*x*) = 720*x*

and so on:

*f *^{(6)}(*x*) = 720.

By the time we've taken as many derivatives as the degree of the original polynomial, we've *used up* all the *x*'s, and we're left with a constant. If we take another derivative we're taking the derivative of a constant, so

*f *^{(7)}(*x*) = 0

If we take any further derivatives we'll be taking the derivative of the function 0, so every derivative from here on will be zero.

Let *f* (*x*) = sin *x*. Since 10 and 50 differ by 40, which is a multiple of 4,

*f* ^{(10)}(*x*) = *f* ^{(50)(x).}

### Sample Problem

Let *f* (*x*) = sin *x*. Since 10 and 50 differ by 40, which is a multiple of 4,

*f* ^{(10)}(*x*) = *f* ^{(50)(x).}

Let *f* (*x*) = sin *x*. Since 10 and 50 differ by 40, which is a multiple of 4,

*f* ^{(10)}(*x*) = *f* ^{(50)(x).}

We've been writing *f* (*x*) for the original function. We could also write

*f* ^{(0)}(*x*),

since taking the "zero-th" derivative of the function means the same thing as not taking any derivatives at all.

We say a function *f* is *infinitely differentiable* if *f* ^{(n)} exists for all whole numbers *n*.