Extending Differentiation

This chapter is about differentiating composite functions. When you have completed it, you should
1. be able to differentiate composite function of the form f(F(x))
2. be able to apply differentiation to rates of change, and to related rates of change.


Differentiating (ax + b)^n

Find dy/dx for
(a) y = (2x + 1)^3
(b) y = (1 - 3x)^4








(c) y = √(3x + 2)

(d) y = 1/ 1-2x





Recall that :
dy/dx is the rate at which y changes with respect to x,


dy/du is the rate at which y changes with respect to u,



du/dx is the rate at which u changes with respect to x,


so :


Related rates of change

You often need to calculate the rate at which one quantity varies with another when one of them is time, then rate of change of r with respect to time t is dr/dt.

Questions like this can be answered by using the chain rule. The situation is best described by a problem.

Ex. Suppose that spherical balloon is being inflated at a constant rate of 5m3s-1. At a particular moment, the radius of the balloon is 4metres. Find how fast the radius of the balloon is increasing at that instant.

First translate the information into a mathematical form. Let V m3 be the volume of the balloon, and let r metres be is radius. Let t seconds be the time for which the balloon has been inflating. Then you are given that dv/dt = 5 and r = 4, and you are asked to find dr/dt at that moment.

Your other piece of info is that the balloon is spherical , so that V = 4/3πr^3 .

Using chain rule :









You can now use 




Substituting the various values into the chain rule formula gives






Therefore, rearranging this equation, you can find that the answer is :

In practice you do not need to write down so much detail.

More exercises will be uploaded soon!

Posted in: composite function,differentiation,rates of change