Let y be the expression you want to differentiate:
[tex]y=\log_7(5-3x) \implies 7^y=5-3x[/tex]
Now,
[tex]7^y=e^{\ln(7^y)}=e^{y\ln(7)}[/tex]
Use the chain rule to differentiate both sides with respect to x :
[tex]\ln(7)e^{y\ln(7)}\dfrac{\mathrm dy}{\mathrm dx}=-3[/tex]
Solve for dy/dx :
[tex]\ln(7)7^y\dfrac{\mathrm dy}{\mathrm dx}=-3[/tex]
[tex]\dfrac{\mathrm dy}{\mathrm dx}=-\dfrac3{\ln(7)7^y}[/tex]
[tex]\dfrac{\mathrm dy}{\mathrm dx}=\boxed{-\dfrac3{\ln(7)(5-3x)}}[/tex]
