Ahh, finally a calculus question :D
When you are deriving two functions that are multiplied by each other, you need to use the product rule. I'll rewrite the definition here:
(I would just write f prime and g prime but there is no coma in the equation tool on this app. This is a better version:)
(d/dx) f(x)g(x) = f(x)g'(x) + f '(x)g(x)
So you have e^x and cos(x) multiplied together and you need to derive the whole function?
The first part of the derivative is the first function times the derivative of the second. Im going to pick cos(x) as my first function and e^x as the second one but it really doesn't matter which one you chose first.
With that being said, the first part looks like this:
(Remember that the derivative of e^x is e^x)
Now, we just have to add the second function times the derivative of the first function, which in this case equals negative sin times e^x
(Rememebr that the derivative of cos(x) is equal to -sin(x)
And that is your derivative ;)