Suppose that we have two functions f(x) and g(x) and they are both differentiable.

1. We define f(x)=(f.g)(x)then the derivative of F(x) is

F'(x)=f'(g(x))g(x)

2. If we have y=f(u) and u=g(x) then the derivative of y is,

dy/dx = (dy/du)*(du/dx)

### General Tips

1. Direct Proportion:

Two quantities are said to be directly proportional, if on the increase (or decrease) of the one, the other increases (or decreases) to the same extent.

Eg. Cost is directly proportional to the number of articles.
(More Articles, More Cost)

2. Indirect Proportion:

Two quantities are said to be indirectly proportional, if on the increase of the one, the orther decreases to the same extent and vice-versa.

Eg. The time taken by a car is covering a certain distance is inversely proportional to the speed of the car. (More speed, Less is the time taken to cover a distance.)

Note: In solving problems by chain rule, we compare every item with the term to be found out.

Example 1:

A fort had provision of food for 150 men for 45 days. After 10 days, 25 men left the fort. The number of days for which the remaining food will last, is:

Solution:

After 10 days : 150 men had food for 35 days.

Suppose 125 men had food for x days.

Now, Less men, More days (Indirect Proportion)

125 : 150 :: 35 : x

125 x x = 150 x 35

x = (150 x 35)/125

x = 42.

Example 2:

A man completes 5/8 of a job in 10 days. At this rate, how many more days will it takes him to finish the job?
Solution:

Work done = 5/8

Balance work = (1-5/8)=3/8

Let the required number of days be x.

Then, 5/8 : 3/8 :: 10 : x

(5/8)*x = (3/8)*10

x = (3/8)*(8/5)*10

x = 6.

Example 3:

A wheel that has 6 cogs is meshed with a larger wheel of 14 cogs. When the smaller wheel has made 21 revolutions, then the number of revolutions mad by the larger wheel is:

Solution

Let the required number of revolutions made by larger wheel be x.

Then, More cogs, Less revolutions (Indirect Proportion)

=> 14 : 6 :: 21 : x

=> 14 x x = 6 x 21

=> (6 *21)/14
x = 9.