find f'(x) if f(x)=a^x

is in undefined, xe, of e^x?

To find the derivative of f(x) = a^x, where a is a constant, we can use the properties of exponential functions and logarithms.

First, we rewrite f(x) as f(x) = e^(ln(a^x)) since e^(ln(x)) is equal to x.

Next, using the property of logarithms, we can bring down the exponent in the natural logarithm:

f(x) = e^(x * ln(a)).

Now, we can differentiate using the chain rule. The derivative of e^(x * ln(a)) with respect to x is:

f'(x) = (d/dx) e^(x * ln(a)) = ln(a) * e^(x * ln(a)).

Therefore, the derivative of f(x) = a^x with respect to x is f'(x) = ln(a) * a^x.

To answer the second part of your question, since the derivative involves the natural logarithm ln(a), it is only defined when a is a positive constant. If a is negative or zero, the expression ln(a) is undefined, and therefore the derivative f'(x) is also undefined.