If `(d)/(dx)(int_(0)^(y)e^(-t^(2))dt+int_(0)^(x^(2)) sin^(2) tdt)=0, "find" (dy)/(dx).`

Answer

To solve the given problem, we need to differentiate the expression involving definite integrals and find \( \frac{dy}{dx} \).

**Step 1: Write down the equation to differentiate.**
We start with the equation given in the problem:
\[ \frac{d}{dx} \left( \int_0^y e^{-t^2} dt + \int_0^{x^2} \sin^2 t \, dt \right) = 0 \]

**Step 2: Apply the Fundamental Theorem of Calculus.**
Using the Fundamental Theorem of Calculus and the chain rule, we differentiate each integral:
\[ \frac{d}{dx} \left( \int_0^y e^{-t^2} dt \right) = e^{-y^2} \cdot \frac{dy}{dx} \]
\[ \frac{d}{dx} \left( \int_0^{x^2} \sin^2 t \, dt \right) = \sin^2(x^2) \cdot \frac{d}{dx}(x^2) = \sin^2(x^2) \cdot 2x \]

**Step 3: Combine the derivatives.**
Now we can combine these results:
\[ e^{-y^2} \cdot \frac{dy}{dx} + 2x \sin^2(x^2) = 0 \]

**Step 4: Solve for \( \frac{dy}{dx} \).**
Rearranging the equation to isolate \( \frac{dy}{dx} \):
\[ e^{-y^2} \cdot \frac{dy}{dx} = -2x \sin^2(x^2) \]
\[ \frac{dy}{dx} = -2x \sin^2(x^2) e^{y^2} \]

**Final Result:**
Thus, the final answer is:
\[ \frac{dy}{dx} = -2x e^{y^2} \sin^2(x^2) \]

---