Find the equation(s) of the tangent line(s) to the graph of the indicated equation at the point(s) with the given value of .

Find the equation(s) of the tangent line(s) to the graph of the given equation at .
(Type an equation. Use a comma to separate answers as needed.)

Ask by Bartlett Cook. in the United States

Mar 24,2025

Question

Find the equation(s) of the tangent line(s) to the graph of the indicated equation at the point(s) with the given value of .

Find the equation(s) of the tangent line(s) to the graph of the given equation at .
(Type an equation. Use a comma to separate answers as needed.)

Ask by Bartlett Cook. in the United States

Mar 24,2025

UpStudy AI · Accepted Answer

Given the equation

$$
xy - x - 18 = 0
$$

and the value $$ x = 3 $$, we first find the corresponding $$ y $$-coordinate.

1. Substitute $$ x = 3 $$ into the equation:

$$
3y - 3 - 18 = 0
$$

2. Simplify and solve for $$ y $$:

$$
3y - 21 = 0 \quad \Rightarrow \quad 3y = 21 \quad \Rightarrow \quad y = 7
$$

The point of tangency is $$ (3, 7) $$.

3. Differentiate the equation implicitly with respect to $$ x $$. Starting with

$$
xy - x - 18 = 0,
$$

differentiate term by term:

$$
\frac{d}{dx}(xy) - \frac{d}{dx}(x) - \frac{d}{dx}(18) = 0.
$$

Using the product rule for $$ xy $$:

$$
y + x\frac{dy}{dx} - 1 = 0.
$$

4. Solve for $$\frac{dy}{dx}$$:

$$
x\frac{dy}{dx} = 1 - y \quad \Rightarrow \quad \frac{dy}{dx} = \frac{1 - y}{x}.
$$

5. Evaluate the derivative at the point $$ (3, 7) $$:

$$
\frac{dy}{dx}\Big|_{(3,7)} = \frac{1 - 7}{3} = \frac{-6}{3} = -2.
$$

This is the slope of the tangent line.

6. Use the point-slope form of a line, $$ y - y_1 = m(x - x_1) $$, with $$ m = -2 $$ and $$ (x_1, y_1) = (3, 7) $$:

$$
y - 7 = -2(x - 3).
$$

7. Simplify to get the equation of the tangent line:

$$
y - 7 = -2x + 6 \quad \Rightarrow \quad y = -2x + 13.
$$

The equation of the tangent line is

$$
y = -2x + 13.
$$
The equation of the tangent line at $$ x = 3 $$ is $$ y = -2x + 13 $$.