Mary Karanja
Find the equation of the tangent line to the curve at
Equation

Ask by Harper Reyes. in the United States

Jan 22,2025

Question

Mary Karanja Find the equation of the tangent line to the curve $$ y=\frac{1}{x} $$ at $$ \left(6, \frac{1}{6}\right) $$ Equation

UpStudy AI · Accepted Answer

To find the equation of the tangent line to the curve $$ y = \frac{1}{x} $$ at the point $$ \left(6, \frac{1}{6}\right) $$, follow these steps:

### Step 1: Find the Derivative
The derivative of a function gives the slope of the tangent line at any point on the curve.

Given:
$$ y = \frac{1}{x} $$

Rewrite $$ y $$ as:
$$ y = x^{-1} $$

Differentiate with respect to $$ x $$:
$$ \frac{dy}{dx} = -x^{-2} = -\frac{1}{x^2} $$

### Step 2: Determine the Slope at $$ x = 6 $$
Evaluate the derivative at $$ x = 6 $$ to find the slope ($$ m $$) of the tangent line at that point.

$$ m = \left. \frac{dy}{dx} \right|_{x=6} = -\frac{1}{6^2} = -\frac{1}{36} $$

### Step 3: Use the Point-Slope Formula
The point-slope form of a line is:
$$ y - y_1 = m(x - x_1) $$
where $$ (x_1, y_1) $$ is the point of tangency.

Given the point $$ \left(6, \frac{1}{6}\right) $$ and $$ m = -\frac{1}{36} $$, plug these into the formula:
$$ y - \frac{1}{6} = -\frac{1}{36}(x - 6) $$

### Step 4: Simplify to Slope-Intercept Form
Expand and simplify the equation to get it into the slope-intercept form $$ y = mx + b $$:

$$
\begin{align*}
y - \frac{1}{6} &= -\frac{1}{36}x + \frac{6}{36} \
y - \frac{1}{6} &= -\frac{1}{36}x + \frac{1}{6} \
y &= -\frac{1}{36}x + \frac{1}{6} + \frac{1}{6} \
y &= -\frac{1}{36}x + \frac{2}{6} \
y &= -\frac{1}{36}x + \frac{1}{3}
\end{align*}
$$

### Final Equation
The equation of the tangent line is:
$$ y = -\frac{1}{36}x + \frac{1}{3} $$

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**Final Answer:**  
After simplifying, an equation for the tangent line is

$$ y = -\frac{1}{36}\,x + \frac{1}{3} $$
The equation of the tangent line is $$ y = -\frac{1}{36}x + \frac{1}{3} $$.

Mary Karanja
Find the equation of the tangent line to the curve at
Equation

Upstudy AI Solution

Answer

Solution

Step 1: Find the Derivative

Step 2: Determine the Slope at

Step 3: Use the Point-Slope Formula

Step 4: Simplify to Slope-Intercept Form

Final Equation

Beyond the Answer

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