# 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

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.

Didn't find what you are looking for on this page? Try search:

### Online Algebra Solver

This algebra solver can solve a wide range of math problems. (Please be patient while it loads.)

Play a math game.

(Well, not really a math game, but each game was made using math...)

Sign up for the free IntMath Newsletter. Get math study tips, information, news and updates each fortnight. Join thousands of satisfied students, teachers and parents!

Given name: * required

Family name:

email: * required

See the Interactive Mathematics spam guarantee.

### Calculus Lessons on DVD

Easy to understand calculus lessons on DVD. See samples before you commit.