Find the derivative of e^(2x)*sinx using the product rule and chain rule.

We calculate the first derivative of e^(2x)*sinx using the product rule and chain rule.

We begin by using prime notation to write an in-between step: this is really useful if you feel like you're doing too much at once! So we apply the product rule to write down (e^2x)'sinx+e^2x*(sinx)'. Now we compute the derivatives, and the first one requires that we apply the chain rule because e^2x is a function composition. so we get 2e^(2x)*sinx+e^(2x)*cosx. This is a fine way to leave the answer, but we can make it look a little nicer by factoring out e^2x to obtain e^2x*(2sinx+cosx).