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We have \[f(x)=\log_{2} (13x)\] notice that the base here is 2, we can't directly differentiate it. We need to have the base of log as e. We know the logarithm's property \[\log_{a} b=\frac{\log_c{b}}{\log_{c} a}\] so here we have \[f(x)= \frac{\log_e(13x)}{\log_e {2}}\] Now we can write as \[f(x)=\frac{\ln (13x)}{\ln 2}\] Let's differentiate this with respect to x \[f'(x)=\frac{d}{dx}(\frac{\ln (13x)}{\ln 2})\] We get \[f'(x)=\frac{1}{\ln 2}\frac{d}{dx}{\ln (13x)}=\frac{1}{2} \frac{1}{13x} \frac{d}{dx} (13x)\] We get \[f'(x)=\frac{1}{\ln 2}\frac{1}{13x} \times {3}\] We get finally \[f'(x)=\frac{3}{\ln 2}\frac{1}{3x1}\]
 2 years ago
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