Answer
$$\frac{dy}{dx} = (x-2)^{x+1}\left({\frac{x+1}{x-2}+\ln(x-2)}\right)$$
Work Step by Step
Logarithmic Differentiation is a method used to take the derivative of functions utlizing the logarithmic derivative of a function. Logarithmss were invented by John Napier in order to make mathematics easier, as it does with logarithmic differentiation:
$$\ y = (x-2)^{x+1}$$
$$\ln y = (x+1) \ln (x-2)$$
$$\frac{1}{y} (\frac{dy}{dx}) = (x+1)(\frac{1}{x-2}) + \ln(x-2)$$
$$\frac{dy}{dx} = y[\frac{x+1}{x-2} + \ln(x-2)]$$
$$ = \frac{dy}{dx} = (x-2)^{x+1}({\frac{x+1}{x-2}+\ln(x-2)})$$
