# How to Convert this Derivative?

• I
I have a derivative of a function with respect to ##\log \left(r\right)##:

\begin{equation*}
\frac{dN\left(r\right)}{d \log\left(r\right)} = \frac{N}{\sqrt{2\pi} \log\left(\sigma\right)} \exp\left\{-\frac{\left[\log \left(r\right) - \log\left(r_M\right)\right]^2}{2 \left[\log\left(\sigma\right)\right]^2}\right\}
\end{equation*}

I need to know the derivative of this function with respect to ##r##, that is ##\frac{dN\left(r\right)}{dr}##, how shall I do this? I was told that I just need to multiply the function with the derivative of the logarithm, that is

\begin{equation*}
\frac{dN\left(r\right)}{dr} = \frac{dN\left(r\right)}{d \log\left(r\right)} \cdot \frac{d \log\left(r\right)}{dr}
\end{equation*}

Is this correct? Even though ##N\left(r\right)## is a function of ##\log\left(r\right)##?

Delta2
Homework Helper
Gold Member
Yes this is correct, it is a well known theorem for the derivative of the composition of two functions (in your case the two functions are N(y) and y(r)=logr)), known as chain rule for derivatives.

https://en.wikipedia.org/wiki/Chain_rule

Ssnow
Gold Member
Yes, its correct, it is the chain rule in one variable.

Alright. Thank you for your help!

• Ssnow