Finding the Derivative of e^x: Step by Step Explanation for Differentiating the Exponential Function

Derivative of e^x

The derivative of e^x can be found using the basic rules of differentiation

The derivative of e^x can be found using the basic rules of differentiation. Let’s proceed step by step:

Step 1: Recall the definition of the derivative. The derivative of a function f(x) can be defined as the limit of the difference quotient as h approaches 0:
f'(x) = lim(h->0) [ f(x + h) – f(x) ] / h

Step 2: Apply the definition of the derivative to e^x. We have:
f(x) = e^x
f'(x) = lim(h->0) [ e^(x + h) – e^x ] / h

Step 3: Simplify the expression inside the limit. Using the properties of exponents, we can rewrite the numerator as:
f'(x) = lim(h->0) [ e^x * e^h – e^x ] / h
= lim(h->0) [ e^x * (e^h – 1) ] / h

Step 4: Factor out e^x from the numerator:
f'(x) = e^x * lim(h->0) [ e^h – 1 ] / h

Step 5: Now, we can evaluate the limit. As h approaches 0, e^h also approaches 1. Therefore:
lim(h->0) [ e^h – 1 ] / h = 1

Step 6: Substitute the limit back into the expression:
f'(x) = e^x * 1
= e^x

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

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