Derivative of e^x
e^x
The derivative of e^x with respect to x is simply e^x. This means that if we have a function f(x) = e^x, then f'(x) = d/dx(e^x) = e^x.
To see why this is true, we can use the definition of the derivative:
f'(x) = lim(h→0) [f(x+h) – f(x)] / h
Substituting f(x) = e^x, we get:
f'(x) = lim(h→0) [e^(x+h) – e^x] / h
We can then use the rules of exponentiation to simplify the expression in the numerator:
f'(x) = lim(h→0) [e^x * e^h – e^x] / h
= lim(h→0) [e^x * (e^h – 1)] / h
= e^x * lim(h→0) (e^h – 1) / h
Now, we recognize that the limit in the last expression is the definition of the number e, so we can substitute it to obtain:
f'(x) = e^x * e
= e^x
So, the derivative of e^x with respect to x is e^x.
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