Find (dy)/(dx) at (-6,1) if xy=(6^y)/(x+5) . (Rc

To find \(\frac{dy}{dx}\) at \((-6, 1)\) for the equation \(xy = \frac{6^y}{x+5}\), we need to implicitly differentiate the given equation with respect to \(x\).<br /><br />Given:<br />\[ xy = \frac{6^y}{x+5} \]<br /><br />First, let's rewrite the equation for clarity:<br />\[ xy = \frac{6^y}{x+5} \]<br /><br />Now, differentiate both sides with respect to \(x\):<br /><br />\[ \frac{d}{dx} \left( xy \right) = \frac{d}{dx} \left( \frac{6^y}{x+5} \right) \]<br /><br />Using the product rule on the left side:<br />\[ y + x \frac{dy}{dx} = \frac{d}{dx} \left( \frac{6^y}{x+5} \right) \]<br /><br />Now, apply the quotient rule to the right side:<br />\[ y + x \frac{dy}{dx} = \frac{(x+5) \cdot \frac{d}{dx}(6^y) - 6^y \cdot \frac{d}{dx}(x+5)}{(x+5)^2} \]<br /><br />Next, differentiate \(6^y\) with respect to \(x\):<br />\[ \frac{d}{dx}(6^y) = 6^y \ln(6) \cdot \frac{dy}{dx} \]<br /><br />And differentiate \(x+5\) with respect to \(x\):<br />\[ \frac{d}{dx}(x+5) = 1 \]<br /><br />Substitute these into the equation:<br />\[ y + x \frac{dy}{dx} = \frac{(x+5) \cdot 6^y \ln(6) \cdot \frac{dy}{dx} - 6^y \cdot 1}{(x+5)^2} \]<br /><br />Simplify the right side:<br />\[ y + x \frac{dy}{dx} = \frac{6^y \ln(6) \cdot \frac{dy}{dx} (x+5) - 6^y}{(x+5)^2} \]<br /><br />Factor out \(\frac{dy}{dx}\) on the right side:<br />\[ y + x \frac{dy}{dx} = \frac{6^y \ln(6) \cdot \frac{dy}{dx} (x+5) - 6^y}{(x+5)^2} \]<br /><br />Combine terms:<br />\[ y + x \frac{dy}{dx} = \frac{6^y \ln(6) \cdot \frac{dy}{dx} (x+5) - 6^y}{(x+5)^2} \]<br /><br />Factor out \(\frac{dy}{dx}\) on the right side:<br />\[ y + x \frac{dy}{dx} = \frac{6^y \ln(6) \cdot \frac{dy}{dx} (x+5) - 6^y}{(x+5)^2} \]<br /><br />Combine terms:<br />\[ y + x \frac{dy}{dx} = \frac{6^y \ln(6) \cdot \frac{dy}{dx} (x+5) - 6^y}{(x+5)^2} \]<br /><br />Factor out \(\frac{dy}{dx}\) on the right side:<br />\[ y + x \frac{dy}{dx} = \frac{6^y \ln(6) \cdot \frac{dy}{dx} (x+5) - 6^y}{(x+5)^2} \]<br /><br />Combine terms:<br />\[ y + x \frac{dy}{dx} = \frac{6^y \ln(6) \cdot \frac{dy}{dx} (x+5) - 6^y}{(x+5)^2} \]<br /><br />Factor out \(\frac{dy}{dx}\) on the right side:<br />\[ y + x \frac{dy}{dx} = \frac{6^y \ln(6) \cdot \frac{dy}{dx} (x+5) - 6^y}{(x+5)^2} \]<br /><br />Combine terms:<br />\[ y + x \frac{dy}{dx} = \frac{6^y \ln(6) \cdot \frac{dy}{dx} (x+5) - 6^y}{(x+5)^2} \]<br /><br />Factor out \(\frac{dy}{dx}\) on the right side:<br />\[ y + x \frac{dy}{dx} = \frac{6^y \ln(6) \cdot \frac{dy}{dx} (x+5) - 6^y}{(x+