Differentiate the function y = e^x.

The derivative of y = e^x is y' = e^x.

To differentiate y = e^x, we use the power rule of differentiation. The power rule states that if y = x^n, then y' = nx^(n-1). In this case, we have y = e^x, which can be written as y = (e)^x. Using the power rule, we get:

y' = d/dx (e^x)
= e^x

Therefore, the derivative of y = e^x is y' = e^x. This means that the slope of the tangent line to the graph of y = e^x at any point (x, e^x) is e^x. The graph of y = e^x is an exponential function that increases rapidly as x increases. The value of e^x approaches infinity as x approaches infinity, and the value of e^x approaches 0 as x approaches negative infinity.

The function y = e^x is used in many areas of mathematics and science, including calculus, probability theory, and physics. It is also used in finance and economics to model exponential growth and decay.