Let's look into differentiation and the different rules in A-Level Maths!
Chain rule:
The chain rule is used to differentiate composite functions y=f(g(x)) (i.e. functions of a function)
The chain rule operates by making the substitution, u=g(x). Then
![broken image](http://custom-images.strikinglycdn.com/res/hrscywv4p/image/upload/c_limit,fl_lossy,h_9000,w_1200,f_auto,q_auto/4979856/987723_687979.jpeg)
Example: Differentiate
![broken image](http://custom-images.strikinglycdn.com/res/hrscywv4p/image/upload/c_limit,fl_lossy,h_9000,w_1200,f_auto,q_auto/4979856/360329_185140.jpeg)
![broken image](http://custom-images.strikinglycdn.com/res/hrscywv4p/image/upload/c_limit,fl_lossy,h_9000,w_1200,f_auto,q_auto/4979856/539698_911252.jpeg)
Product rule
![broken image](http://custom-images.strikinglycdn.com/res/hrscywv4p/image/upload/c_limit,fl_lossy,h_9000,w_1200,f_auto,q_auto/4979856/221851_872014.jpeg)
It is used when there is a product of 2 functions.
Example:
Differentiate y= (3x2 - 2x)e1-2x
This is a product of 2 functions:
u = 3x2 - 2x ⇨ du/dx = 6x - 2
v= e1-2x ⇨ dv/dx= - 2e1-2x
Using the productrule formula:
![broken image](http://custom-images.strikinglycdn.com/res/hrscywv4p/image/upload/c_limit,fl_lossy,h_9000,w_1200,f_auto,q_auto/4979856/373771_265073.jpeg)
Quotient rule
![broken image](http://custom-images.strikinglycdn.com/res/hrscywv4p/image/upload/c_limit,fl_lossy,h_9000,w_1200,f_auto,q_auto/4979856/736366_998734.jpeg)
It is used for differentiating quotients (i.e.fractions).
Example: Differentiate
![broken image](http://custom-images.strikinglycdn.com/res/hrscywv4p/image/upload/c_limit,fl_lossy,h_9000,w_1200,f_auto,q_auto/4979856/462376_53238.jpeg)
![broken image](http://custom-images.strikinglycdn.com/res/hrscywv4p/image/upload/c_limit,fl_lossy,h_9000,w_1200,f_auto,q_auto/4979856/385418_708141.jpeg)
Factorise top in order to simplify:
![broken image](http://custom-images.strikinglycdn.com/res/hrscywv4p/image/upload/c_limit,fl_lossy,h_9000,w_1200,f_auto,q_auto/4979856/480129_490789.jpeg)
Remember these results:
![broken image](http://custom-images.strikinglycdn.com/res/hrscywv4p/image/upload/c_limit,fl_lossy,h_9000,w_1200,f_auto,q_auto/4979856/819221_407436.jpeg)
Finding the derivative when x = f(y):
We use the result
![broken image](http://custom-images.strikinglycdn.com/res/hrscywv4p/image/upload/c_limit,fl_lossy,h_9000,w_1200,f_auto,q_auto/4979856/317461_65294.jpeg)
Example:
If x = 4y - y2, find dy/dx in terms of y.
![broken image](http://custom-images.strikinglycdn.com/res/hrscywv4p/image/upload/c_limit,fl_lossy,h_9000,w_1200,f_auto,q_auto/4979856/219032_861921.jpeg)
Connected rates of change:
The word rate is associated with differentiation.
Key steps:
(1) Interpret mathematically the information given in the question.
(2) Write down the relevant formula connecting the variables and differentiate it.
(3) Decide on what quantity needs to be calculated.
(4) Use the chain rule to calculate it.
Example:
The volume of a sphere is decreasing at a rate of 20cm3/s. Find the rate at which the radius is increasing when the radius is 3 cm.
Solution:
From question, dV/dt = -20
Connecting formula:
![broken image](http://custom-images.strikinglycdn.com/res/hrscywv4p/image/upload/c_limit,fl_lossy,h_9000,w_1200,f_auto,q_auto/4979856/965837_680288.jpeg)
We want dr/dt
using chain rule
![broken image](http://custom-images.strikinglycdn.com/res/hrscywv4p/image/upload/c_limit,fl_lossy,h_9000,w_1200,f_auto,q_auto/4979856/7107_412372.jpeg)
![broken image](http://custom-images.strikinglycdn.com/res/hrscywv4p/image/upload/c_limit,fl_lossy,h_9000,w_1200,f_auto,q_auto/4979856/460968_397152.jpeg)
Drafted by Eunice (Maths)