How do you express the derivative of a^x?

The derivative of the function ( a^x ) can be derived using the rules of differentiation, particularly applying the chain rule.

When differentiating ( a^x ), you treat it in the context of exponential functions. Precisely, the expression ( a^x ) can be rewritten using the natural logarithm as ( e^{x \ln(a)} ). This transformation is useful because it allows the application of the known derivative of the exponential function.

Using the chain rule, the derivative of ( e^{u} ) (where ( u = x \ln(a) )) is ( e^{u} \cdot \frac{du}{dx} ). Here, the derivative of ( u = x \ln(a) ) with respect to ( x ) is simply ( \ln(a) ). Therefore, we have:

[

\frac{d}{dx}(a^x) = \frac{d}{dx}(e^{x \ln(a)}) = e^{x \ln(a)} \cdot \ln(a) = a^x \ln(a).

]

Thus, the derivative of ( a^x ) is indeed expressed as ( a^x \ln