1. Intro :
chain rule
It is the way to find the derivative
when u have a complicated
equation that is a function inside
a larger, outer function.
2. Now for example
Find dy/dx ( or “ find the
derivative”)
• Y=(3x+1)^7
You will need the chain rule
when you have a smaller X
expression or function
inside a larger function.
3. So the chain rule is little cryptic as
to what to do
That’s why it has some another
name
Outside-inside rule
𝐷𝑒𝑟𝑖𝑣𝑎𝑡𝑖𝑣𝑒 𝑜𝑓 𝑜𝑢𝑡𝑠𝑖𝑑𝑒 𝑓𝑢𝑛𝑐𝑡𝑖𝑜𝑛
, 𝑙𝑒𝑎𝑣𝑖𝑛𝑔 𝑖𝑛𝑠𝑖𝑑𝑒 𝑎𝑙𝑜𝑛𝑒
.
𝐷𝑒𝑟𝑖𝑣𝑎𝑡𝑖𝑣𝑒 𝑜𝑓
𝑖𝑛𝑠𝑖𝑑𝑒 𝑓𝑢𝑛𝑐𝑡𝑖𝑜𝑛
4. Lets try that example now
Y=(3x+1)^7
Dy/dx or y’= 7( 3x+1)^6.(3)
inner left alone and we took the derivative
of inside expression because it is not only x
Dy/dx= 21(3x+1)^6
5. How do we know that we need Chain rule for derivative
Y=(3x+1)^7 for this we just did definitely the chain rule
Here comes some more functions
Will we need chain rule for them
1- 3x+1 ( No coz there is no outer function)
2- x^7 (its not composite function too
3- (3x+1)^7 ( definitely we will need because it has inside function
4- (x+1)^7 (yes)
5- (x^2+1)^7 (yes)
6. Lets do another one :
h(x)=(x^2+5x-6)^9
h’(x) or dh/dx= 9(x^2+5x-6)^8.(2x+5)
Let do chain rule or no chain rule:
X^2+5x-6 (no)
X^9 (no)
(X-6)^9 (yes)
(x^2+5x-6)^9 (yes)
7. 3: for Trig
y= sin(x^2-3x)
Dy/dx= cos(x^2-3x).(2x-3)
Lets do chain no chain
SinX (NO)
Sin 3X (yes coz 3x is the inside function)
Sin X^2(yes X^2 is inside function for sine)
