If f(x) = e^x, what is equal to f'(e)?

Derivatives are one of the most important concepts in calculus which have many applications in various other fields. 

Answer: If f(x) = e^x, then f'(e) = e^e.

Let's understand the solution in detail.

Explanation:

We use the first principle formula of derivatives to calculate the result.

Hence, by first principle of derivatives, we have:

f'(x) = lim_h→0 { f(x + h) - f(x) } / h

Hence,

f'(e) = lim_h→0 { f(e + h) - f(e) } / h

= lim_h→0 { e^{e + h} - e^e } / h

= lim_h→0 [ e^e (e^h - 1) ] / h

Now, we know that lim_h→0 { (e^h - 1) / h } = 1, We can also verify this by the L'Hopital's rule.

Hence, f'(e) = e^e.

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Hence, if f(x) = e^x, then f'(e) = e^e.