Product Rule Differentiation (Single Variable)

Calculus Tutorial

Product Rule Differentiation (Single Variable)

Intro

The product rule is used when you are finding the derivative of a function that has one dependent variable expression multiplied by another dependent variable expression. Compare this function:

f(x) = x2 + 2x5

…with this function:

g(x) = x4cos(x)

Function g(x) can be thought of as two separate functions multiplied together:

h(x) = x4
j(x) = cos(x)

g(x) = h(x)j(x) = x4cos(x)

That is why taking the derivative of g(x) means taking the derivative of the product of more than one function, which is why we will use the product rule.

Sample Problem

Find the derivative of f(x):

f(x) = x3(ln(x))

Solution

Remember that a function like f(x) is thought of as more than one function multiplied together. The product rule is as follows given a function f(x):

f(x) = g(x)h(x)
f'(x) = g'(x)h(x) + g(x)+h'(x)

In this example, our g(x) and h(x) are defined when there are two distinct expressions of the dependent variable; in other words, we cannot reduce or combine x3 and ln(x) together, so these can be considered two different functions of x. Following the formula, we can break f(x) like so:

f(x) = x3(ln(x)) = g(x)h(x)
g(x) = x3
h(x) = ln(x)

Now that we’ve defined the components of f(x), we can plug in g(x) and h(x) into the product rule, using their respective derivatives:

g'(x) = 3x2
h'(x) = 1/x

f'(x) = 3x2ln(x) + x3(1/x)

We can reduce and combine terms further to get the final answer:

f'(x) = 3x2ln(x) + x2
f'(x) = x2(3ln(x) + 1)



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