Questions: Find the derivative of y with respect to x. y = ln (1-x)/(x+5)^6

Solution Steps

To find the derivative of \( y \) with respect to \( x \) for the given function \( y = \ln \frac{1-x}{(x+5)^6} \), we can use the properties of logarithms and the chain rule. First, simplify the logarithmic expression using the properties of logarithms, then differentiate the resulting expression.

Given the function: \[ y = \ln \left( \frac{1-x}{(x+5)^6} \right) \] We can use the properties of logarithms to simplify: \[ y = \ln(1-x) - \ln((x+5)^6) \] \[ y = \ln(1-x) - 6\ln(x+5) \]

Differentiate \( y \) with respect to \( x \): \[ \frac{dy}{dx} = \frac{d}{dx} \left( \ln(1-x) - 6\ln(x+5) \right) \] Using the chain rule: \[ \frac{dy}{dx} = \frac{d}{dx} \ln(1-x) - 6 \frac{d}{dx} \ln(x+5) \] \[ \frac{dy}{dx} = \frac{-1}{1-x} - 6 \cdot \frac{1}{x+5} \]

Combine the terms: \[ \frac{dy}{dx} = \frac{-1}{1-x} - \frac{6}{x+5} \] Find a common denominator: \[ \frac{dy}{dx} = \frac{-(x+5) - 6(1-x)}{(1-x)(x+5)} \] Simplify the numerator: \[ \frac{dy}{dx} = \frac{-x-5 - 6 + 6x}{(1-x)(x+5)} \] \[ \frac{dy}{dx} = \frac{5x - 11}{(1-x)(x+5)} \]

Final Answer