# Second Derivatives and Beyond

### Example 1

Let *f *(*x*) = *x*^{3} + 2.

Find *f *^{2}(*x*).

### Example 2

Let *f *(*x*) = *x*^{3} + 2.

Find *f *^{(2)}(*x*).

### Example 3

Suppose *f* (*x*) is a degree 3 polynomial.

a. What is the degree of *f *'(*x*)?

b. What is the degree of *f* ^{(2)}(*x*) ?

c. What is the degree of *f* ^{(3)}(*x*) ?

### Example 4

Suppose *f* (*x*) is a degree 9 polynomial.

a. What is the degree of *f* ^{8}(*x*) ?

b. What is the degree of *f* ^{9}(*x*)?

c. What is the degree of *f* ^{10}(*x*)?

### Example 5

Suppose *f* (*x*) is a degree *n* polynomial.

a. What is the degree of *f* ^{(n - 1)}(*x*)?

b. What is the degree of *f* ^{(n)}(*x*)?

c. What is the degree of *f* ^{(k)}(*x*) where *k* is any integer greater than *n*?

### Example 6

Exercise. Find the first nine derivatives of the function *f* (*x*) = sin *x*.

### Example 7

Let *f* (*x*) = sin *x*. Find the derivative.

*f* ^{(12)}(*x*)

### Example 8

Let *f* (*x*) = sin *x*. Find the derivative.

*f* ^{(22)}(*x*)

### Example 9

Let *f* (*x*) = sin *x*. Find the derivative.

*f* ^{(33)}(*x*)

### Example 10

What are some functions that are infinitely differentiable?