# 9. Higher Derivatives

by M. Bourne

We can continue to find the derivatives of a derivative. We find the

• second derivative by taking the derivative of the first derivative,
• third derivative by taking the derivative of the second derivative... etc

### Example 1

If y=x^5+3x^3-2x+7, then what are the higher derivatives?

## Application - Acceleration

We saw before that acceleration is the rate of change of velocity:

a=(dv)/(dt)

But we also know that velocity is the rate of change of displacement:

v=(ds)/(dt)

So it follows that the second derivative of displacement will give us acceleration:

a=(d^2s)/(dt^2)

### Example 2

If the displacement (in metres) at time t (in seconds) of an object is given by

s = 4t3 + 7t2 − 2t,

find the acceleration at time t = 10.

## Higher Derivatives of Implicit Functions

### Example 3

(The Answers for these two questions contain short video explanations.)

a. Find the second derivative of the implicit function xy + y2 = 4.

b. Find the value of the second derivative of the implicit function in part (a) when x = 2, where y > 0.