We are tasked with finding the equations of all tangent lines to the curve $-3 + x^2 = xy$ when $x = 3$.

Step 1: Implicit Differentiation

The given equation is:

$-3 + x^2 = xy$

We'll differentiate both sides with respect to $x$, keeping in mind that $y$ is a function of $x$ (so we must apply the product and chain rules).

The derivative of the left-hand side:

$\frac{d}{dx} \left( -3 + x^2 \right) = 2x$

Now for the right-hand side, using the product rule on $xy$:

$\frac{d}{dx}(xy) = x \frac{dy}{dx} + y$

Thus, the implicit differentiation gives:

$2x = x \frac{dy}{dx} + y$

Step 2: Solve for $\frac{dy}{dx}$

Now, isolate $\frac{dy}{dx}$:

$2x - y = x \frac{dy}{dx}$

$\frac{dy}{dx} = \frac{2x - y}{x}$

Step 3: Find $y$ when $x = 3$

Substitute $x = 3$ into the original equation to find $y$:

$-3 + 3^2 = 3y$

$-3 + 9 = 3y$

$6 = 3y$

$y = 2$

Step 4: Find the Slope of the Tangent Line

Substitute $x = 3$ and $y = 2$ into the derivative formula:

$\frac{dy}{dx} = \frac{2(3) - 2}{3} = \frac{6 - 2}{3} = \frac{4}{3}$

Thus, the slope of the tangent line at $x = 3$ is $\frac{4}{3}$.

Step 5: Equation of the Tangent Line

We now use the point-slope form of the equation of a line:

$y - y_1 = m(x - x_1)$

where $m = \frac{4}{3}$, $x_1 = 3$, and $y_1 = 2$. Substituting these values:

$y - 2 = \frac{4}{3}(x - 3)$

Simplifying:

$y - 2 = \frac{4}{3}x - 4$

$y = \frac{4}{3}x - 2$

Thus, the equation of the tangent line at $x = 3$ is:

$y = \frac{4}{3}x - 2$

Would you like more details, or do you have any questions?

5 Related Questions:

1. How do you handle implicit differentiation with multiple variables?
2. What is the significance of the point-slope form in finding tangent lines?
3. How do you find the tangent line to a curve if the equation is not easily solvable for $y$?
4. Can there be more than one tangent line at a given $x$ value for different types of curves?
5. How does implicit differentiation differ from explicit differentiation in complex functions?

Tip:

Always double-check your derivative rules when working with implicit differentiation, especially with products of functions like $xy$.