Product Rule
Neil Trivedi
Teacher
Contents
Product Rule
We use the product rule to differentiate expressions where two functions are multiplied together.
Product Rule
Note: You may have seen NeilDoesMaths think of it like this in his tutorials:
(Differentiate term)( term) (Differentiate term)( term)
Example 1:
Differentiate with respect to
a)
Single Step: We have and . Differentiate using the product rule.
We have and .
b)
Single Step: We have and . Differentiate using the product rule.
We have and (using the chain rule).
Often, we will set to equal to find turning points, so we should practise factorising completely. We always factorise the smallest power. In this case, that will be and . Then, we divide each term by both, subtracting the powers accordingly. You can always expand again to check if you have factorised properly.
No answer provided.
Example 2:
Differentiate with respect to
a)
Single Step: We have and . Differentiate using the product rule.
We have and .
b)
Single Step: We have and . Differentiate using the product rule.
We have and .
No answer provided.
Example 3:
A curve has equation .
a) Find .
Single Step: We have and . Differentiate using the product rule.
We have and (using the chain rule).
Factorising common factors from each term, we have common factors of and .
Notice we can factorise out as well. We get:
b) Find the gradient of the curve at the point where .
Single Step: Substitute into to get the gradient.
c) Find the coordinates of the turning points of the curve.
Step 1: The turning points of the curve occur when . Work out the coordinates of these points.
Divide both sides by , then equate each bracket to and solve for .
Step 2: Find the corresponding coordinates by substituting the coordinates into the original equation.
Therefore, the coordinates of the turning points are and .
No answer provided.
Practice Questions