How do you differentiate a function containing e?

y = e^x is an interesting function because for any value of x, the corresponding value of y is always equal to the gradient of the curve at that point.Therefore, f(x) is equivalent to f'(x) - the derivative of the function. When you are met with slightly more complex functions, such as y = e^{2x^2}, you can find the derivative of the function by following a simple rule:If f(x) = e^g(x), then f'(x) = g'(x)e^g(x). Therefore, the power, to which e is raised, remains the same and the function of e is multiplied by the derivative of the power.If we return to our example of y = e^{2x^2}, we know that our g(x) = 2x², so g'(x) = 4x.From this we know that f'(x) = (e^{2x^2}) x (4x) = 4xe^{2x^2}.