Differentiation Product Rule

When you have a function consisting of a pair of functions that multiply together, then the product rule helps to differentiate it. One function is designated f(x), whilst the other is designated g(x). When the function splits in this way, the product rule shows how to perform the differentiation operation on them.

Product Rule Example

If you had a simple function y = (x+1)x ^ -1 you can use the product rule to find its derivative. This is about as simple as I can make it for the beginner without being too obvious.

You can split the function so that f(x) = x + 1, and g(x) = x ^ -1. Each half of the function has an x in it.

If you wanted to make yourself look clever, you could rewrite the function like this and call yourself a genius.

The next stage is to differentiate the individual functions. As you can see, d/dx f(x) = 1, and d/dx g(x) = -x ^ -2, as it is just a simple differentiation of a power.

Then all that remains is to substitute these values into the product rule formula shown at the top of this page. As you can see, it is a simple matter of multiplying out the brackets and very basic algebra.

These are just the final steps after simplifying a little. As you can see the product rule works extremely well to yield a result.

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