d/dx [e^x]
e^x
The derivative of e^x with respect to x is e^x.
To show this, we can use the definition of a derivative:
d/dx [e^x] = limit as h approaches 0 of [e^(x+h) – e^x] / h
Using the properties of exponents, we can simplify e^(x+h) as e^x * e^h. Substituting this in the expression above, we get:
limit as h approaches 0 of [e^x * e^h – e^x] / h
We can factor out e^x from the numerator to get:
limit as h approaches 0 of e^x [e^h – 1] / h
Now, we can apply L’Hôpital’s rule to evaluate this limit:
limit as h approaches 0 of e^x [e^h – 1] / h = limit as h approaches 0 of e^x * e^h / 1 = e^x * 1
Therefore, the derivative of e^x with respect to x is e^x.
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