The document introduces the quotient rule for taking the derivative of functions that are divided. Specifically, it states that if h(x) is defined as u(x)/v(x), then the quotient rule says that h'(x) is equal to (v(x)u'(x) - u(x)v'(x))/(v(x))^2. This rule, which rearranges the fraction and applies the product rule, was developed by Maria Gaetana Agnesi in her 1748 textbook as a way to help her brothers learn algebra. It then provides two examples of using the quotient rule to find the gradient function.

1. MHL Quotient Rule 1
The Quotient Rule
We know how to find the derivative of the product of two
functions but how do we find the derivative when tow functions are divided?
This approach to the quotient rule is credited to Maria Gaetana Agnesi (1718 –
1799) who wrote the first known mathematics textbook Analytical
Institutions (1748) to help her brothers learn algebra.
Let’s have a recap of the product rule:
Now let
𝒉( 𝒙) =
𝒖(𝒙)
𝒗(𝒙)
Rearrange this to make u(x) the subject
Now using the product rule find the derivative of 𝒖(𝒙)
Now rearrange to make 𝒉′(𝒙) the subject
Low Dhi minus Hi Dlow draw a line and square below!
The Product Rule:
The Quotient Rule:

2. MHL Quotient Rule 2
Find the gradient function for:
1) 𝑦 =
1+3𝑥
𝑥2+1
2) 𝑦 =
√ 𝑥
(1−2𝑥)2