Understanding the Derivative of the Exponential Function e^x | Step-by-Step Explanation and Application of the Chain Rule

d/dx e^x

The mathematical expression “d/dx e^x” represents the derivative of the exponential function e^x with respect to x

The mathematical expression “d/dx e^x” represents the derivative of the exponential function e^x with respect to x. To find the derivative of e^x, we can apply differentiation rules.

The derivative of e^x can be found using the chain rule. Let’s break it down step by step:

Step 1: Identify the function and variable
In this case, the function is e^x, and the variable of differentiation is x.

Step 2: Apply the chain rule
The chain rule states that if we have a composition of functions, f(g(x)), the derivative of that composition can be found by multiplying the derivative of the outer function f'(g(x)) and the derivative of the inner function g'(x).

In our case, the outer function is the exponential function e^x, and the inner function is x.

The derivative of the exponential function e^x is itself since it is a special function that remains unchanged under differentiation. In other words, d/dx (e^x) = e^x.

The derivative of x with respect to x is simply 1.

Step 3: Apply the chain rule formula
Using the chain rule, we can multiply the derivatives of the outer and inner functions:

d/dx (e^x) = (e^x) * (d/dx(x))
=> d/dx (e^x) = e^x

Therefore, the derivative of e^x with respect to x is simply e^x.

In summary, d/dx (e^x) = e^x.

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