Questions: Use implicit differentiation to find d y/d x. ln (x y) + 6 x = 15

Solution Steps

To find \(\frac{dy}{dx}\) using implicit differentiation for the given equation \(\ln(xy) + 6x = 15\):

1. Differentiate both sides of the equation with respect to \(x\).
2. Apply the product rule to the term \(\ln(xy)\).
3. Solve for \(\frac{dy}{dx}\).

Given the equation: \[ \ln(xy) + 6x = 15 \]

Differentiate both sides with respect to \(x\): \[ \frac{d}{dx} \left( \ln(xy) + 6x \right) = \frac{d}{dx} (15) \]

For the term \(\ln(xy)\), use the chain rule and product rule: \[ \frac{d}{dx} \ln(xy) = \frac{1}{xy} \cdot \frac{d}{dx} (xy) = \frac{1}{xy} \left( y + x \frac{dy}{dx} \right) \]

Differentiate \(6x\) and the constant \(15\): \[ \frac{d}{dx} (6x) = 6 \] \[ \frac{d}{dx} (15) = 0 \]

Combine the differentiated terms: \[ \frac{1}{xy} \left( y + x \frac{dy}{dx} \right) + 6 = 0 \]

Multiply through by \(xy\) to clear the fraction: \[ y + x \frac{dy}{dx} + 6xy = 0 \]

Isolate \(\frac{dy}{dx}\): \[ x \frac{dy}{dx} = -y - 6xy \] \[ \frac{dy}{dx} = \frac{-y - 6xy}{x} \] \[ \frac{dy}{dx} = -\frac{y(1 + 6x)}{x} \]

Final Answer