1. The Chain Rule

2. What is to be learned?
How to differentiate a composite function

3. f(x) = (x + 5)2
= x2 + 10x + 25 
f(x) = (x + 5)7

4.  y = (2x + 5)7 remember sin(2x + 30) = 0.8 dy/dx sinA = 0.8,
where A = 2x + 30
let y= u7
where u = 2x + 5
dy/du
= 7u6
du/dx
= 2
= dydu
dy/du X
du/dx
dudx
dy/du
du/dx =
dy/dx
= 7u6 2
dy/dx
= 14u6

5.  y = (2x + 5)7 remember sin(2x + 30) = 0.8 dy/dx sinA = 0.8,
where A = 2x + 30
let y= u7
where u = 2x + 5
dy/du
= 7u6
du/dx
= 2
= dydu
dy/du
du/dx
dudx
dy/du
du/dx =
dy/dx
= 7u6 2
dy/dx
= 14u6

6.  y = (2x + 5)7 remember sin(2x + 30) = 0.8 dy/dx sinA = 0.8,
where A = 2x + 30
let y= u7
where u = 2x + 5
dy/du
= 7u6
du/dx
= 2
= dydu
dy/du
du/dx
dudx
dy/du
du/dx =
dy/dx
= 7u6 2
dy/dx
The Chain Rule
= 14u6
= 14(2x+5)6

7. y = (7x + 2)5
let y= u5
where u = 7x + 2
dy/du
= 5u4
du/dx
= 7
dy/dx =
dy/du
du/dx
= 5u4 7
dy/dx
= 35u4
= 35(7x + 2)4

8. The Chain Rule For Type (ax + b)n
Key to the chain rule
dy/dx =
dy/du
du/dx

9. y = (x2 + 2)9
let y= u9
where u = x2 + 2 *
dy/du
= 9u8
du/dx
= 2x
dy/dx =
dy/du
du/dx
= 9u8 2x
dy/dx
= 18x u8
= 18x(x2 + 2)8
from*

10. Fast Tracking
y = (7x + 2)5
let y= u5
where u = 7x + 2
dy/du
= 5u4
du/dx
= 7
dy/dx =
dy/du
du/dx
=5(7x + 2)4
= 5u4 7
dy/dx
derivative of bit in brackets