Question

Find the derivative of y = ㏒5
`((2x+1)(x-3))`

Answer 1

Given

`y = \log_5\bigg((2x + 1) (x - 3)\bigg)`

Expand the logarithm on the right side. (change-of-base and product-to-log identities)

`y = (\ln(2x + 1) + \ln(x - 3))/(\ln(5))`

Now differentiate both sides with respect to
`x`.

`y' = \frac1{\ln(5)} \bigg(\ln(2x+1)\bigg)' + \frac1{\ln(5)} \bigg(\ln(x-3)\bigg)'`

`y' = ((2x+1)')/(\ln(5)\,(2x+1)) + ((x-3)')/(\ln(5)\,(x-3))`

`y' = (2)/(\ln(5)\,(2x+1)) + (1)/(\ln(5)\,(x-3))`

`y' = \boxed{(4x-5)/(\ln(5)\,(2x+1)(x-3))}`