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I am trying to understand the derivative of the function $G(d)=a^d$. I am told it is the following (lecturers comments on my work)
$$ \frac{\partial}{\partial d}(a^d) = \frac{\partial}{\partial d}(e^{d\log a}) = (\log a)a^d $$
I'm not sure where the exponential comes from. Looking at this site the rule is
$$ \frac{\partial}{\partial d}(a^d) = a^d \ln a $$
and converting from $\ln$ to $\log$ does not seem to help. Can someone point me in the right direction?
The issue is whether the base is the variable or the exponent is the variable. If the base was the variable, i.e., $d^a$ then the derivative would be $a \, d^{a-1}$, applying power rule.
If the variable $d$ is in the exponent, you are dealing with an exponential function. Derivative of $e^x$ is $e^x$, then combining with chain rule, you get the result your lecturer gives.
For details, we want to have an equivalent expression for $a^d$ with $e$ as the base (so that we can use the derivative of $e^x$). If we temporarily let $y=a^d$, taking natural log we get $\ln y = d \, \ln a$, and exponentiating back you get $y = e^{d \, \ln a}$. Now taking the derivative with respect to $d$ (applying derivative of $e^x$ along with chain rule) you get $\frac{dy}{dd} = e^{d \, \ln a} \, \ln a = a^d \, \ln a$
HTH.
