Product Rule

This is a simple guide about how to take the derivative of a product of two functions

This post will use the notation f '(x) to show the first derivative of f with respect to x

To find the derivative of a product, simply use the rule and formula shown below.

Given a function h(x) = f(x)*g(x):

The derivative h'(x) = f(x)*g'(x) + g(x)*f '(x)

Examples

1. Y = xsinx
1. Recall that h'(x) = f(x)*g'(x) + g(x)*f '(x)
2. In this case, f(x) = x and g(x) = sinx
3. Therefore f ‘(x) = 1 and g’(x) = cosx
4. Substituting into the product rule formula, the answer dy/dx = x*cosx + 1*sinx
5. dy/dx = xcosx + sinx

2. Y = (x^3)lnx
1. Recall that h'(x) = f(x)*g'(x) + g(x)*f '(x)
2. In this case, f(x) = x^3 and g(x) = lnx
3. Therefore f ‘(x) = 3x^2 and g’(x) = 1/x
4. Substituting into the formula the answer dy/dx = (x^3)*(1/x) + (lnx)*(3x^2)
5. dy/dx = x^2 + (3x^2)lnx

3. Y = (sinx)(cosx)
1. Recall that h'(x) = f(x)*g'(x) + g(x)*f '(x)
2. In this case, f(x) = sinx and g(x) = cosx
3. Therefore f ‘(x) = cosx and g’(x) = -sinx
4. Substituting into the formula the answer dy/dx = sinx(-sinx) + (cosx)(cosx)
5. dy/dx = cos^2(x) - sin^2(x) or cos2x

Similar to the product rule, there is a rule known as the quotient rule. Information about the quotient rule can

be found at the site linked below http://tutorial.math.lamar.edu/Classes/CalcI/ProductQuotientRule.aspx