How would I differentiate y=2(e^x)sin(5x) ?
We can see that we have the product of two different functions, and so we need to use the product rule. We will separate the function and label each part u and v for clarity. So we let u = 2ex and let v = sin(5x), where y =uv we now recall the product rule: d(uv)/dx = ud(v)/dx + vd(u)/dx now we simply differentiate u and v separately and plug them into our formula, remembering how to differentiate exponentials and using the chain rule to differentiate sin(5x). du/dx = 2ex and dv/dx = 5cos(5x) so we now have dy/dx = (2ex)(5cos(5x)) + (sin(5x))(2ex) to simplify and make our answer look a bit nicer, we could take out a factor of 2ex to get 2ex(5cos(5x) + sin(5x))
