Discovering the Derivative of e^x Using the Chain Rule: A Comprehensive Explanation

Derivitive of e^x

The derivative of the function f(x) = e^x, denoted as f'(x) or dy/dx, can be found by applying the chain rule of differentiation

The derivative of the function f(x) = e^x, denoted as f'(x) or dy/dx, can be found by applying the chain rule of differentiation.

Let’s begin by writing the given function in the form y = e^x. Now, we need to find dy/dx (the derivative of y with respect to x).

Using the chain rule, we know that if y = u^n, then dy/dx = n * u^(n-1) * du/dx, where u represents a function of x and n is a constant.

In this case, u = e and n = 1. Therefore, we have y = e^1 = e.

Applying the chain rule, we find dy/dx as follows:

dy/dx = n * u^(n-1) * du/dx
= 1 * e^(1-1) * du/dx
= 1 * e^0 * du/dx
= 1 * 1 * du/dx
= du/dx

Since du/dx is the derivative of u = e^x, we can conclude that dy/dx = du/dx.

Thus, the derivative of e^x is e^x.

In summary, the derivative of the function f(x) = e^x is f'(x) = e^x.

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