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.
More Answers:
[next_post_link]