Derivative of $$$\frac{\ln\left(a\right)}{\ln\left(b\right)}$$$ with respect to $$$a$$$
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Find $$$\frac{d}{da} \left(\frac{\ln\left(a\right)}{\ln\left(b\right)}\right)$$$.
Solution
Apply the constant multiple rule $$$\frac{d}{da} \left(c f{\left(a \right)}\right) = c \frac{d}{da} \left(f{\left(a \right)}\right)$$$ with $$$c = \frac{1}{\ln\left(b\right)}$$$ and $$$f{\left(a \right)} = \ln\left(a\right)$$$:
$${\color{red}\left(\frac{d}{da} \left(\frac{\ln\left(a\right)}{\ln\left(b\right)}\right)\right)} = {\color{red}\left(\frac{\frac{d}{da} \left(\ln\left(a\right)\right)}{\ln\left(b\right)}\right)}$$The derivative of the natural logarithm is $$$\frac{d}{da} \left(\ln\left(a\right)\right) = \frac{1}{a}$$$:
$$\frac{{\color{red}\left(\frac{d}{da} \left(\ln\left(a\right)\right)\right)}}{\ln\left(b\right)} = \frac{{\color{red}\left(\frac{1}{a}\right)}}{\ln\left(b\right)}$$Thus, $$$\frac{d}{da} \left(\frac{\ln\left(a\right)}{\ln\left(b\right)}\right) = \frac{1}{a \ln\left(b\right)}$$$.
Answer
$$$\frac{d}{da} \left(\frac{\ln\left(a\right)}{\ln\left(b\right)}\right) = \frac{1}{a \ln\left(b\right)}$$$A