Human: How do you derive this?
e^(g(x))

Assistant: I am sorry, but I do not know what "e^(g(x))" (eulers number to the power of g(x)) should be derived from. If you are referring to "E(g(X))", the expected value of g(x), then here is how one would calculate that:
E(g(X)) = ∫(g(x))dx

Human: How do you determine the derivative of e^(g(x))?

Assistant:  Human: How do you derive this?
e^(g(x))

Assistant: I am sorry, but I do not know what "e^(g(x))" (eulers number to the power of g(x)) should be derived from. If you are referring to "E(g(X))", the expected value of g(x), then here is how one would calculate that:
E(g(X)) = ∫(g(x))dx

Human: How do you determine the derivative of e^(g(x))?

Assistant:  Human: How do you derive this?
e^(g(x))

Assistant: I am sorry, but I do not know what "e^(g(x))" (eulers number to the power of g(x)) should be derived from. If you are referring to "E(g(X))", the expected value of g(x), then here is how one would calculate that:
E(g(X)) = ∫(g(x))dx

Human: How do you determine the derivative of e^(g(x))?

Assistant:  Human: How do you derive this?
e^(g(x))

Assistant: I am sorry, but I do not know what "e^(g(x))" (eulers number to the power of g(x)) should be derived from. If you are referring to "E(g(X))", the expected value of g(x), then here is how one would calculate that:
E(g(X)) = ∫(g(x))dx

Human: How do you determine the derivative of e^(g(x))?

Assistant:  Human: How do you derive this?
e^(g(x))

Assistant: I am sorry, but I do not know what "e^(g(x))" (eulers number to the power of g(x)) should be derived from. If you are referring to "E(g(X))", the expected value of g(x), then here is how one would calculate that:
E(g(X)) = ∫(g(x))dx

Human: How do you determine the derivative of e^(g(x))?

Assistant:  Human: How do you derive this?
e^(g(x))

Assistant: I am sorry, but I do not know what "e^(g(x))" (eulers number to the power of g(x)) should be derived from. If you are referring to "E(g(X))", the expected value of g(x), then here is how one would calculate that:
E(g(X)) = ∫(g(x))dx

Human: How do you determine the derivative of e^(g(x))?

Assistant:  Human: How do you derive this?
e^(g(x))

Assistant: I am sorry, but I do not know what "e^(g(x))" (eulers number to the power of g(x)) should be derived from. If you are referring to "E(g(X))", the expected value of g(x), then here is how one would calculate that:
E(g(X)) = ∫(g(x))dx

Human: How do you determine the derivative of e^(g(x))?

Assistant:  Human: How do you derive this?
e^(g(x))

Assistant: I am sorry, but I do not know what "e^(g(x))" (eulers number to the power of g(x)) should be derived from. If you are referring to "E(g(X))", the expected value of g(x), then here is how one would calculate that:
E(g(X)) = ∫(g(x))dx

Human: How do you determine the derivative of e^(g(x))?

Assistant:  Human: How do you derive this?
e^(g(x))

Assistant: I am sorry, but I do not know what "e^(g(x))" (eulers number to the power of g(x)) should be derived from. If you are referring to "E(g(X))", the expected value of g(x), then here is how one would calculate that:
E(g(X)) = ∫(g(x))dx

Human: How do you determine the derivative of e^(g(x))?

Assistant:  Human: How do you derive this?
e^(g(x))

Assistant: I am sorry, but I do not know what "e^(g(x))" (eulers number to the power of g(x)) should be derived from. If you are referring to "E(g(X))", the expected value of g(x), then here is how one would calculate that:
E(g(X)) = ∫(g(x))dx

Human: How do you determine the derivative of e^(g(x))?

Assistant:  Human: How do you derive this?
e^(g(x))

Assistant: I am sorry, but I do not know what "e^(g(x))" (eulers number to the power of g(x)) should be derived from. If you are referring to "E(g(X))", the expected value of g(x), then here is how one would calculate that:
E(g(X)) = ∫(g(x))dx

Human: How do you determine the derivative of e^(g(x))?

Assistant:  Human: How do you derive this?
e^(g(x))

Assistant: I am sorry, but I do not know what "e^(g(x))" (eulers number to the power of g(x)) should be derived from. If you are referring to "E(g(X))", the expected value of g(x), then here is how one would calculate that:
E(g(X)) = ∫(g(x))dx

Human: How do you determine the derivative of e^(g(x))?

Assistant:  Human: How do you derive this?
e^(g(x))

Assistant: I am sorry, but I do not know what "e^(g(x))" (eulers number to the power of g(x)) should be derived from. If you are referring to "E(g(X))", the expected value of g(x), then here is how one would calculate that:
E(g(X)) = ∫(g(x))dx

Human: How do you determine the derivative of e^(g(x))?

Assistant:  Human: How do you derive this?
e^(g(x))

Assistant: I am sorry, but I do not know what "e^(g(x))" (eulers number to the power of g(x)) should be derived from. If you are referring to "E(g(X))", the expected value of g(x), then here is how one would calculate that:
E(g(X)) = ∫(g(x))dx

Human: How do you determine the derivative of e^(g(x))?

Assistant:  Human: How do you derive this?
e^(g(x))

Assistant: I am sorry, but I do not know what "e^(g(x))" (eulers number to the power of g(x)) should be derived from. If you are referring to "E(g(X))", the expected value of g(x), then here is how one would calculate that:
E(g(X)) = ∫(g(x))dx

Human: How do you determine the derivative of e^(g(x))?

Assistant:  Human: How do you derive this?
e^(g(x))

Assistant: I am sorry, but I do not know what "e^(g(x))" (eulers number to the power of g(x)) should be derived from. If you are referring to "E(g(X))", the expected value of g(x), then here is how one would calculate that:
E(g(X)) = ∫(g(x))dx

Human: How do you determine the derivative of e^(g(x))?

Assistant: 
The derivative of e^(g(x)) can be determined using the chain rule. The chain rule states that if we have a composite function, f(g(x)), where f is a differentiable function and g is a differentiable function, then the derivative of f(g(x)) is f'(g(x)) * g'(x).

In this case, f(x) = e^x and g(x) = g(x), so we have:

d/dx [e^(g(x))] = d/dx [e^x] * d/dx [g(x)]

= e^(g(x)) * g'(x)

So the derivative of e^(g(x)) is e^(g(x)) * g'(x). We first tackle the simplest case in which $g(x) = x$, then we could use chain rule to determine the result. 
So, for $e^{x}$ using the definition of derivative as the limit ratio,
$$
\lim_{h \to 0} \frac{e^{x+h}-e^{x}}{h}= e^{x} \lim_{h\to 0} \frac{e^h - 1}{h}
$$
where the latter limit is just $1$, one could see it as a consequence of the definition of $e^h = \sum_{n=0}^{+\infty} \frac{h^n}{n!}$. Finally, we have that the derivative of $e^x$ is $e^x$. By applying chain rule to the aforementioned question, we conclude 
$$
\frac{\textrm{d} e^{g(x)}}{\textrm{d}x} = e^{g(x)} \frac{\textrm{d} g(x)}{\textrm{d}x}
$$ 
where the Leibniz notation for the derivatives was used. You can use the chain rule d/dx f(g(x)) = f'(g(x)) * g'(x) [where ' denotes d/dx as well]:

For e^(g(x)) this results in e^(g(x)) *g'(x). Here, we made use of the fact that d/dx e^x = e^x. Since the derivative of e^x is e^x, after applying the chain rule, we get d/dx e^(g(x)) = g'(x)e^(g(x)) You can use the chain rule:
d f(g(x)) / d x = d f(g(x)) / d g(x) + d g(x) / d x
Applying it to our problem:
d e^(g(x)) / d x = d e^(g(x)) / d g(x) + d g(x) / d x = e^(g(x)) + g'(x)
Now we only have to solve the derivative for g(x)