Questions: ln(x^2+y^2-6x)=5x-6 (a) Find dy/dx in terms of x and y:

Transcript text: \[ \ln \left(x^{2}+y^{2}-6 x\right)=5 x-6 \] (a) Find $\frac{\mathrm{d} y}{\mathrm{~d} x}$ in terms of $x$ and $y$ :

Solution

Solution Steps

To find $\frac{\mathrm{d} y}{\mathrm{~d} x}$ in terms of $x$ and $y$, we need to implicitly differentiate both sides of the given equation with respect to $x$. This involves using the chain rule and the product rule where necessary.

Solution Approach

Differentiate both sides of the equation with respect to $x$.
Apply the chain rule to the left-hand side and the product rule to the right-hand side.
Solve for $\frac{\mathrm{d} y}{\mathrm{~d} x}$.

Step 1: Differentiate Both Sides with Respect to $ x $

Given the equation: \[ \ln \left(x^{2}+y^{2}-6 x\right)=5 x-6 \]

We need to find $\frac{\mathrm{d} y}{\mathrm{~d} x}$. To do this, we will differentiate both sides of the equation with respect to $ x $.

Step 2: Apply the Chain Rule to the Left Side

The left side of the equation is $\ln \left(x^{2}+y^{2}-6 x\right)$. Using the chain rule, we get: \[ \frac{\mathrm{d}}{\mathrm{d} x} \left( \ln \left(x^{2}+y^{2}-6 x\right) \right) = \frac{1}{x^{2}+y^{2}-6 x} \cdot \frac{\mathrm{d}}{\mathrm{d} x} \left( x^{2}+y^{2}-6 x \right) \]

Step 3: Differentiate the Argument of the Logarithm

Now, we differentiate $ x^{2}+y^{2}-6 x $ with respect to $ x $: \[ \frac{\mathrm{d}}{\mathrm{d} x} \left( x^{2}+y^{2}-6 x \right) = 2x + 2y \frac{\mathrm{d} y}{\mathrm{~d} x} - 6 \]

Step 4: Differentiate the Right Side

The right side of the equation is $ 5x - 6 $. Differentiating with respect to $ x $, we get: \[ \frac{\mathrm{d}}{\mathrm{d} x} \left( 5x - 6 \right) = 5 \]

Step 5: Combine the Results

Combining the results from Steps 2, 3, and 4, we have: \[ \frac{1}{x^{2}+y^{2}-6 x} \cdot (2x + 2y \frac{\mathrm{d} y}{\mathrm{~d} x} - 6) = 5 \]

Step 6: Solve for $\frac{\mathrm{d} y}{\mathrm{~d} x}$

First, multiply both sides by $ x^{2}+y^{2}-6 x $: \[ 2x + 2y \frac{\mathrm{d} y}{\mathrm{~d} x} - 6 = 5(x^{2}+y^{2}-6 x) \]

Next, isolate the term involving $\frac{\mathrm{d} y}{\mathrm{~d} x}$: \[ 2y \frac{\mathrm{d} y}{\mathrm{~d} x} = 5(x^{2}+y^{2}-6 x) - 2x + 6 \]

Simplify the right side: \[ 2y \frac{\mathrm{d} y}{\mathrm{~d} x} = 5x^{2} + 5y^{2} - 30x - 2x + 6 \] \[ 2y \frac{\mathrm{d} y}{\mathrm{~d} x} = 5x^{2} + 5y^{2} - 32x + 6 \]

Finally, solve for $\frac{\mathrm{d} y}{\mathrm{~d} x}$: \[ \frac{\mathrm{d} y}{\mathrm{~d} x} = \frac{5x^{2} + 5y^{2} - 32x + 6}{2y} \]