# 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= 4t^{3}+ 7t^{2}− 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* + *y*^{2} = 4.

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

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