The product rule provides us with a method for finding the derivative of a product of two functions. The quotient rule provides us with a method for finding the derivative of a quotient of two functions. Before learning either of these two rules we must first have a good understanding of how to find the derivative of the basic functions (f(x) = c, where c is a constant, xn, ln x, ex, sin x and cos x), how to use the multiple rule to find the derivatives of a constant times any of these basic functions, and how to use the sum and difference rules to find the derivatives of the sums or differences of any of these basic functions and their constant multiples.
Let us first consider the derivative of the following product:
Now consider the product of the following derivatives:
From the above we can see the following important point:
To find the derivative of a product we use the following product rule:
In words we say: The derivative of a product is the second times the derivative of the first plus the first times the derivative of the second.
Click here to see a proof of the product rule.
Click on the question marks to see the following example done step-by-step:
There is an easier way to solve the above problem. If we multiply the functions first and then find the derivative we obtain:
You can see that by either method we get the same answer. In many cases, however, the product rule is the only option. For example, the following can only be found using the product rule:
The following visual method can be used as an alternative to memorizing the formula for the product rule:
Find the derivative of
Try solving these examples using the product rule.
Some people find the Product Rule easier to remember if it is written with the first multiplication in the reverse order to that above.
The two ways are totally equivalent. Choose one way of writing it down and stick to that way.