How do you find the derivative of the function e^(2x)?

To find the derivative of the function e^(2x), we will use the chain rule, which is a fundamental technique in calculus for differentiating composite functions.

First, recall that the derivative of the exponential function e^u, where u is a function of x, is given by:

u' * e^u

In our case, u = 2x. Therefore, we first need to find the derivative of u with respect to x.

Calculating this gives:

u' = d(2x)/dx = 2

Now, applying the chain rule:

f'(x) = u' * e^u = 2 * e^(2x)

So, the derivative of the function e^(2x) is:

f'(x) = 2 * e^(2x)

In summary, if you need to differentiate e^(2x), you simply multiply the original function by the derivative of the exponent:

f'(x) = 2 * e^(2x)

This result is useful in various applications, especially when solving problems that involve exponential growth or decay.