The Product Rule
In calculus we use the product rule to find the derivative of two differentiable functions. In this post I will discuss the product rule and give several examples of how to use it to find the derivative of the product of two functions. The statement of the product rule is as follows:
Product Rule:
If f and g are differentiable functions then (f(x)g(x))' = f'(x)g(x) + f(x)g'(x).
The product rule tells us that the derivative of the product of two functions is equal to the derivative of the first function times the second function plus the derivative of the second function times the first function.
For our first example of how to use the product rule suppose that we want to find the derivative of the function h(x) = xex. The function h(x) is the product of the functions f(x) = x and g(x) = ex. It follows from the product rule that h'(x) = (x)'ex + x(ex)' = 1ex + xex = ex + xex.
Now suppose we want to take the derivative of the function h(x) = cos(x)sin(x). The function h(x) is the product of the functions f(x) = cos(x) and g(x) = sin(x). According to the product rule we have h'(x) = (cos(x))'sin(x) + cos(x)(sin(x))' = -sin(x)sin(x) + cos(x)cos(x) = -sin2(x) + cos2(x).
As we become more familiar with the product rule we will not write out all the steps and details as in the above two examples. For our third example suppose that h(x) = x2tan(x). From the product rule we find that h'(x) = 2xtan(x) + x2sec2(x).
In this post we have gone over the product rule and given several examples of how to calculate the derivative of the product of two functions using this rule. It should be clear that the product rule is not limited to taking the derivative of the product of only two functions. We can apply the product rule to find the derivative of a function which is the product of more than two functions. To do this we can apply the product rule multiple times until we have the desired derivative.
References:
https://en.wikipedia.org/wiki/Product_rule
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cool post, you just took me back to school years' math :)
Thank you!
Good explanation.
Thank you!
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Thanks!