The problem involves a definite integral function $F(x)$ and appears to require evaluating $F'(x)$, the derivative of $F(x)$. Let's analyze the given function and its derivative.

The Function:

$F(x) = \int_x^{x^2} e^{t^2} \, dt$

Step-by-Step Solution:

1. Apply the Leibniz Rule for differentiation of an integral with variable limits: $F'(x) = \frac{d}{dx} \int_{a(x)}^{b(x)} f(t) \, dt = f(b(x)) \cdot b'(x) - f(a(x)) \cdot a'(x)$ Here:

   • $a(x) = x$, so $a'(x) = 1$,
   • $b(x) = x^2$, so $b'(x) = 2x$,
   • $f(t) = e^{t^2}$.

2. Substitute the Limits into the Leibniz Rule: $F'(x) = e^{(x^2)^2} \cdot 2x - e^{x^2} \cdot 1$

3. Simplify the expression: $F'(x) = 2x e^{x^4} - e^{x^2}$

Final Answer:

$F'(x) = 2x e^{x^4} - e^{x^2}$

Would you like a detailed explanation of any step or related concepts?

5 Related Questions:

1. What is the Leibniz rule for differentiating integrals with variable limits?
2. How do you compute higher-order derivatives of functions like $F(x)$?
3. What are some examples of definite integrals with variable limits?
4. How does $e^{t^2}$ behave in integrals, given it cannot be expressed in elementary terms?
5. Can this problem be extended to multivariable calculus scenarios?

Tip:

When dealing with integrals with variable limits, always check if the Fundamental Theorem of Calculus or Leibniz Rule applies to simplify differentiation.