Given the polar equation r=1. find `dy/dx`.

Options

A. `cot theta`
B. `tan theta`
C. 0
D. `-cot theta`

Answer

To find \( \frac{dy}{dx} \) for the polar equation \( r = 1 \), we will follow these steps:

### Step 1: Convert the polar equation to Cartesian coordinates
The polar coordinates \( (r, \theta) \) can be converted to Cartesian coordinates \( (x, y) \) using the following relationships:
\[ x = r \cos \theta \]
\[ y = r \sin \theta \]
Given \( r = 1 \), we can substitute this into the equations:
\[ x = 1 \cdot \cos \theta = \cos \theta \]
\[ y = 1 \cdot \sin \theta = \sin \theta \]

### Step 2: Write the Cartesian equation
From the above equations, we can express the relationship between \( x \) and \( y \):
\[ x^2 + y^2 = 1 \]
This is the equation of a circle with radius 1 centered at the origin.

### Step 3: Differentiate the equation with respect to \( x \)
Now we differentiate both sides of the equation \( x^2 + y^2 = 1 \) with respect to \( x \):
\[ \frac{d}{dx}(x^2) + \frac{d}{dx}(y^2) = \frac{d}{dx}(1) \]
Using the chain rule for \( y^2 \):
\[ 2x + 2y \frac{dy}{dx} = 0 \]

### Step 4: Solve for \( \frac{dy}{dx} \)
Rearranging the equation to isolate \( \frac{dy}{dx} \):
\[ 2y \frac{dy}{dx} = -2x \]
\[ \frac{dy}{dx} = -\frac{x}{y} \]

### Step 5: Substitute back the polar coordinates
Since we have \( x = \cos \theta \) and \( y = \sin \theta \), we can substitute these back into our expression for \( \frac{dy}{dx} \):
\[ \frac{dy}{dx} = -\frac{\cos \theta}{\sin \theta} = -\cot \theta \]

### Final Answer
Thus, the value of \( \frac{dy}{dx} \) for the polar equation \( r = 1 \) is:
\[ \frac{dy}{dx} = -\cot \theta \]