x=a cos t, y=b sin t

Answer

To solve the problem where \( x = a \cos t \) and \( y = b \sin t \), we need to find \( \frac{dy}{dx} \).

### Step-by-step Solution:

1. **Differentiate \( x \) with respect to \( t \)**:
\[ x = a \cos t \]
To find \( \frac{dx}{dt} \):
\[ \frac{dx}{dt} = -a \sin t \]

**Hint**: Remember that the derivative of \( \cos t \) is \( -\sin t \).

2. **Differentiate \( y \) with respect to \( t \)**:
\[ y = b \sin t \]
To find \( \frac{dy}{dt} \):
\[ \frac{dy}{dt} = b \cos t \]

**Hint**: Recall that the derivative of \( \sin t \) is \( \cos t \).

3. **Use the chain rule to find \( \frac{dy}{dx} \)**:
The relationship between the derivatives is given by:
\[ \frac{dy}{dx} = \frac{dy/dt}{dx/dt} \]
Substituting the values we found:
\[ \frac{dy}{dx} = \frac{b \cos t}{-a \sin t} \]

**Hint**: Make sure to substitute the derivatives correctly into the chain rule formula.

4. **Simplify the expression**:
\[ \frac{dy}{dx} = -\frac{b}{a} \cdot \frac{\cos t}{\sin t} = -\frac{b}{a} \cot t \]

**Hint**: Remember that \( \cot t = \frac{\cos t}{\sin t} \).

### Final Answer:
\[ \frac{dy}{dx} = -\frac{b}{a} \cot t \]