# Exponential and Logarithm Derivatives

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?

• $\log$ and $\ln$ both denote logarithm to the base $e$. Apr 22, 2017 at 14:34
• And $\log$ is universally used to denote $\log_e$ in mathematics, while $\ln$ is more common where people believe their calculators. Apr 22, 2017 at 14:39
• I dont think that is a universal convention. In many U.S. high school and college textbooks, $\log$ is $\log_{10}$ while $\ln$ corresponds to $\log_e$ Apr 22, 2017 at 14:51
• @Just_to_Answer Sure, but Chappers mentioned "in mathematics"...
– Did
Apr 22, 2017 at 21:42
• @Did Haha...fair enough. Since this discussion has been done quite a bit in math.stackexchange.com/questions/90594/…, I dont think it is worth rehashing. Apr 22, 2017 at 22:42

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 $\frac{dy}{dd} = e^{d \, \ln a} \, \ln a = a^d \, \ln a$