How do you use the product rule?

The product rule is used to find the differential of expressions of the form y = u(x)*v(x) where u(x) and v(x) are functions in terms of x. An example of such an expression could be y = x²sin(x). The product rule states that for y = u(x)*v(x), the first derivative is given by y' = u'(x)v(x) + u(x)v'(x) (the symbol ' refers to the first derivative). Applying this to our example, we first need to define what u(x) and v(x) are. We could let u(x) = x² and v(x) = sin(x). We could have also defined v(x) = x² and u(x) = sin(x). The order in this case doesn't matter as long as one is consistent, but we will be continuing with our first definition. We now need to find what u'(x) and v'(x) are. As u(x) = x², u'(x) = 2x . Also, v(x) = sin(x), v'(x) = cos(x) By applyin the formula y' = u'(x)v(x) + u(x)v'(x) we can therefore find that y' = 2x(sin(x)) + x²(cos(x)).