`Ifint_(pi/3)^xsqrt((3-sin^2t))dt+int_0^ycostdt=0,t h e ne v a l u a t e(dy)/(dx)`

Explanation

To solve the given problem, we need to evaluate the derivative \( \frac{dy}{dx} \) given the equation:

\[ \int_{\frac{\pi}{3}}^{x} \sqrt{3 - \sin^2 t} \, dt + \int_{0}^{y} \cos t \, dt = 0 \]

### Step-by-Step Solution:

1. **Differentiate Both Sides with Respect to \( x \)**:
We apply the Leibniz rule for differentiation under the integral sign. The equation becomes:
\[ \frac{d}{dx} \left( \int_{\frac{\pi}{3}}^{x} \sqrt{3 - \sin^2 t} \, dt \right) + \frac{d}{dx} \left( \int_{0}^{y} \cos t \, dt \right) = 0 \]

2. **Evaluate the First Integral**:
By the Fundamental Theorem of Calculus, we have:
\[ \frac{d}{dx} \left( \int_{\frac{\pi}{3}}^{x} \sqrt{3 - \sin^2 t} \, dt \right) = \sqrt{3 - \sin^2 x} \]

3. **Evaluate the Second Integral**:
For the second integral, we apply the chain rule:
\[ \frac{d}{dx} \left( \int_{0}^{y} \cos t \, dt \right) = \cos y \cdot \frac{dy}{dx} \]

4. **Combine the Results**:
Substituting the results back into the equation gives us:
\[ \sqrt{3 - \sin^2 x} + \cos y \cdot \frac{dy}{dx} = 0 \]

5. **Isolate \( \frac{dy}{dx} \)**:
Rearranging the equation to solve for \( \frac{dy}{dx} \):
\[ \cos y \cdot \frac{dy}{dx} = -\sqrt{3 - \sin^2 x} \]
\[ \frac{dy}{dx} = -\frac{\sqrt{3 - \sin^2 x}}{\cos y} \]

### Final Result:
Thus, the value of \( \frac{dy}{dx} \) is:
\[ \frac{dy}{dx} = -\frac{\sqrt{3 - \sin^2 x}}{\cos y} \]