Third Derivatives and Beyond Examples

Find the third derivative of the function f(x) = 5x⁴ + 4x³ – 7x².

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

f ⁽¹⁾(x) = 20x³ + 12x² – 14x.

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

f ⁽²⁾(x) = 60x² + 24x – 14.

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

f ⁽³⁾(x) = 120x + 24.

Find the fifth derivative of the function f(x) = 5x⁴ + 4x³ – 7x².

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.

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.