Quotient Rule
Neil Trivedi
Teacher
Contents
Quotient Rule
The quotient rule is used when differentiating functions involving the division of two functions.
Quotient Rule
Note: You may have seen NeilDoesMaths express this formula as:
(Differentiate the top term)(Bottom term) (Differentiate the bottom term)(Top term)
(Bottom term)
Example 1:
Find the derivative of:
a)
Single Step: We have and . Differentiate using the quotient rule.
We have and .
b)
Single Step: We have and . Differentiate using the quotient rule.
We have and .
No answer provided.
Example 2:
Show that the derivative of is .
Step 1: Rewrite in terms of and .
We know that (refer to the Reciprocated Trigonometric Functions study note).
Hence, we’re differentiating .
Step 2: We have and . Differentiate using the quotient rule.
We have and .
Use the trigonometric identity to simplify the numerator.
This proves one of our standard results that .
No answer provided.
Example 3:
A curve has the equation .
a) Find an expression for .
Single Step: We have and . Differentiate using the quotient rule.
We have and .
b) Find the coordinates of the turning points of the curve.
Step 1: Find the value of such that .
Multiply through by the denominator to get:
Solving for within the range , we get:
Note: We must be careful here as the question has indicated that , because if we substitute into the denominator of , we would be left with , which is
and we cannot divide by .
Therefore, we only have two solutions and .
Step 2: Find the corresponding coordinates by substituting and into the original equation.
Therefore, the coordinates of the turning points of the curve are and .
c) Show that the tangent to the curve at the point with coordinate has equation
Step 1: Find the corresponding coordinate.
Step 2: Find the gradient of the tangent at the point by substituting into our equation for .
Step 3: Use the formula to find the equation of the tangent.
We have and .
Multiply every term by to get:
No answer provided.
Practice Questions