The Chain Rule | Finding the Derivative of f(u) = e^u – Explained Step by Step

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To find the derivative of the function f(u) = e^u, where u is a variable, we can use the chain rule

To find the derivative of the function f(u) = e^u, where u is a variable, we can use the chain rule. The chain rule states that if we have a composition of functions, then the derivative of the composition is the derivative of the outer function multiplied by the derivative of the inner function.

In this case, the outer function is the exponential function e^x, and the inner function is u. Let’s start by finding the derivative of the outer function.

The derivative of e^x with respect to x is simply e^x. Now, let’s substitute the inner function u back into our derivative:

d/dx (e^u) = d/du (e^u) * du/dx

Since u is a variable in terms of x, du/dx represents the derivative of u with respect to x. In this case, since u is independent of x, du/dx is simply 1.

Therefore, the derivative of f(u) = e^u with respect to x is:

d/dx (e^u) = d/du (e^u) * du/dx = e^u * 1 = e^u

So, the derivative of e^u with respect to x is e^u.

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