find dy/dx if x=a(cos$$ \theta+\theta\sin\theta),y=a\left(\sin\theta-\cos\theta\right)_{} $$

To find $\frac{dy}{dx}$ for the given parametric equations:

\[
x = a(\cos\theta + \theta\sin\theta)
\]
\[
y = a(\sin\theta - \theta\cos\theta)
\]

we will differentiate $x$ and $y$ with respect to $\theta$ first, and then use the formula for parametric differentiation:

\[
\frac{dy}{dx} = \frac{\frac{dy}{d\theta}}{\frac{dx}{d\theta}}
\]

### Step 1: Differentiate $x$ with respect to $\theta$.
\[
x = a(\cos\theta + \theta\sin\theta)
\]
Differentiating with respect to $\theta$:
\[
\frac{dx}{d\theta} = a\left(-\sin\theta + \sin\theta + \theta\cos\theta\right)
\]
Simplifying:
\[
\frac{dx}{d\theta} = a\theta\cos\theta
\]

### Step 2: Differentiate $y$ with respect to $\theta$.
\[
y = a(\sin\theta - \theta\cos\theta)
\]
Differentiating with respect to $\theta$:
\[
\frac{dy}{d\theta} = a\left(\cos\theta - (\cos\theta - \theta\sin\theta)\right)
\]
Simplifying:
\[
\frac{dy}{d\theta} = a\theta\sin\theta
\]

### Step 3: Find $\frac{dy}{dx}$.
Now, use the formula:
\[
\frac{dy}{dx} = \frac{\frac{dy}{d\theta}}{\frac{dx}{d\theta}} = \frac{a\theta\sin\theta}{a\theta\cos\theta}
\]
Simplifying:
\[
\frac{dy}{dx} = \frac{\sin\theta}{\cos\theta} = \tan\theta
\]

Thus, the derivative $\frac{dy}{dx}$ is $\tan\theta$.

### Do you want details or have any questions?

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### 5 Relative Questions:
1. How would the result change if the parametric equations were altered slightly, such as switching the roles of sine and cosine?
2. What happens to $\frac{dy}{dx}$ if $\theta = \frac{\pi}{2}$?
3. Can you express the second derivative $\frac{d^2y}{dx^2}$ in terms of $\theta$?
4. How do these parametric equations describe the curve in Cartesian coordinates?
5. How can you verify this result by converting the parametric equations to a Cartesian equation?

### Tip:
For parametric equations, always differentiate both $x$ and $y$ with respect to the parameter before finding $\frac{dy}{dx}$.

Learn how to find dy/dx for the parametric equations x = a(cos theta + theta sin theta) and y = a(sin theta - theta cos theta) using parametric differentiation. Detailed step-by-step solution provided.

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