How to Use The Product Rule (Differentiation)

A quick guide to using the product rule whilst differentiating.

In this article it will be assumed that you can:

  • Differentiate simple terms,
  • Use the chain rule to differentiate.

In differentiation, the product rule is a combination of the the chain rule and further differentiation.

The product rule must be used when two different terms are attached by multiplication in an equation.

The equation for the product rule is u(Δv/Δx)+v(Δu/Δx), where u is the first term, Δu/Δx is the differential of the first term, v is the second term and Δv/Δx is the differential of the second term.

Let’s take the following example:

y = 2x(3x-1)³

We must split this up into 2 terms by separating them where they are multiplied, so u = 2x and v = (3x-1)³.

We must now differentiate these terms, so Δu/Δx = 2 and Δv/Δx = 9(3x-1)².

Using the formula we can find Δy/Δx, so u(Δv/Δx) = 18x(3x-1)² and v(Δu/Δx) = 2(3x-1)³.

In order to add u(Δv/Δx) and v(Δu/Δx) we must find like terms. The first terms are 18x and 2, and 2 goes into both of these, so we take 2 out and we are left with 9x.

In the second set of terms we have (3x-1)² and 2(3x-1)³, and (3x-1)² goes into both of these, so we must take that out and we are left with 2(3x-1) or 6x-2.

We now have 2, (3x-1)² and the leftovers of 9x and 6x-2. We must add the leftovers together to make 15x-2.

Now we must add all of these terms together and we are finally left with:

Δy/Δx = 2(15x-2)(3x-1)²

Congratulations!

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