log_{7 }(5 - 3x) Find the derivative

Question

log_{7 }(5 - 3x) Find the derivative​

mit · Answer

Let y be the expression you want to differentiate:

$y=\log_7(5-3x) \implies 7^y=5-3x$

Now,

$7^y=e^{\ln(7^y)}=e^{y\ln(7)}$

Use the chain rule to differentiate both sides with respect to x :

$\ln(7)e^{y\ln(7)}\dfrac{\mathrm dy}{\mathrm dx}=-3$

Solve for dy/dx :

$\ln(7)7^y\dfrac{\mathrm dy}{\mathrm dx}=-3$

$\dfrac{\mathrm dy}{\mathrm dx}=-\dfrac3{\ln(7)7^y}$

$\dfrac{\mathrm dy}{\mathrm dx}=\boxed{-\dfrac3{\ln(7)(5-3x)}}$