Schwartz Sullivan
06/12/2023 · Middle School
3. Determine \( d y / d x \), where \( x=a(\theta+\sin \theta) \) and \( y=a \) \( (1-\cos \theta) \)
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Tutor-Verified Answer
Step-by-step Solution
To find \( \frac{dy}{dx} \), we will use the chain rule and implicit differentiation. We have the equations:
\[
x = a(\theta + \sin \theta)
\]
\[
y = a(1 - \cos \theta)
\]
First, we need to find \( \frac{dx}{d\theta} \) and \( \frac{dy}{d\theta} \).
### Step 1: Differentiate \( x \) with respect to \( \theta \)
\[
\frac{dx}{d\theta} = a\left(1 + \cos \theta\right)
\]
### Step 2: Differentiate \( y \) with respect to \( \theta \)
\[
\frac{dy}{d\theta} = a \sin \theta
\]
### Step 3: Use the chain rule to find \( \frac{dy}{dx} \)
Using the chain rule, we have:
\[
\frac{dy}{dx} = \frac{dy/d\theta}{dx/d\theta}
\]
Substituting the derivatives we found:
\[
\frac{dy}{dx} = \frac{a \sin \theta}{a(1 + \cos \theta)}
\]
The \( a \) cancels out:
\[
\frac{dy}{dx} = \frac{\sin \theta}{1 + \cos \theta}
\]
Thus, the final result is:
\[
\frac{dy}{dx} = \frac{\sin \theta}{1 + \cos \theta}
\]
Quick Answer
\[
\frac{dy}{dx} = \frac{\sin \theta}{1 + \cos \theta}
\]
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