Find the third derivative of the function f(x) = 5x^{4} + 4x^{3} - 7x^{2}.

We'll find each derivative along the way. The first derivative is

f ^{(1)}(x) = 20x^{3} + 12x^{2} - 7x.

We take another derivative to find the second derivative of f :

f ^{(2)}(x) = 60x^{2} + 24x - 7.

Finally, one more time to find the third derivative of f:

f ^{(3)}(x) = 120x + 24.

Example 2

Find the fifth derivative of the function f(x) = 5x^{4} + 4x^{3} - 7x^{2}.

We already know the third derivative:

f ^{(3)}(x) = 120x + 24.

We differentiate again:

f ^{(4)}(x) = 120

One more time. Since the derivative of a constant is zero,

f ^{(5)}(x) = 0.

Example 3

Let f (x) = sin x. Find f^{(101)} (x).

We need to know whether 101 differs by a multiple of 4 from 1, 2, 3, or 4. Since 101 is different from 1 by 100, which is a multiple of 4,
f^{(101)}(x) = f^{(1)}(x) = cos x.