If `x = a cos theta` and `y = b sin theta`, find `(dy)/(dx)`

Explanation

To find \(\frac{dy}{dx}\) given the equations \(x = a \cos \theta\) and \(y = b \sin \theta\), we will use the chain rule of differentiation. Here’s a step-by-step solution:

### Step 1: Differentiate \(x\) with respect to \(\theta\)
Given:
\[ x = a \cos \theta \]
To find \(\frac{dx}{d\theta}\), we differentiate \(x\) with respect to \(\theta\):
\[ \frac{dx}{d\theta} = -a \sin \theta \]

### Step 2: Differentiate \(y\) with respect to \(\theta\)
Given:
\[ y = b \sin \theta \]
To find \(\frac{dy}{d\theta}\), we differentiate \(y\) with respect to \(\theta\):
\[ \frac{dy}{d\theta} = b \cos \theta \]

### Step 3: Use the chain rule to find \(\frac{dy}{dx}\)
According to the chain rule:
\[ \frac{dy}{dx} = \frac{dy/d\theta}{dx/d\theta} \]
Substituting the derivatives we found in Steps 1 and 2:
\[ \frac{dy}{dx} = \frac{b \cos \theta}{-a \sin \theta} \]

### Step 4: Simplify the expression
We can simplify the expression:
\[ \frac{dy}{dx} = -\frac{b}{a} \cdot \frac{\cos \theta}{\sin \theta} \]
Recognizing that \(\frac{\cos \theta}{\sin \theta} = \cot \theta\), we can write:
\[ \frac{dy}{dx} = -\frac{b}{a} \cot \theta \]

### Final Answer:
Thus, the final result is:
\[ \frac{dy}{dx} = -\frac{b}{a} \cot \theta \]

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