We are given the problem:

$\frac{d}{dx} \left( \int_5^{x^2} \left(t^2 - 4\right) \, dt \right)$

To solve this, we will use the Leibniz rule for differentiating integrals with variable limits. The rule states:

$\frac{d}{dx} \left( \int_{a(x)}^{b(x)} f(t) \, dt \right) = f(b(x)) \cdot \frac{d}{dx} b(x) - f(a(x)) \cdot \frac{d}{dx} a(x)$

In this case:

• The upper limit of the integral is $x^2$, so $b(x) = x^2$.
• The lower limit is a constant $5$, so $a(x) = 5$.
• The function inside the integral is $f(t) = t^2 - 4$.

Now, applying the Leibniz rule: $\frac{d}{dx} \left( \int_5^{x^2} (t^2 - 4) \, dt \right) = (x^2 - 4) \cdot \frac{d}{dx}(x^2) - (5^2 - 4) \cdot \frac{d}{dx}(5)$ Since $\frac{d}{dx}(x^2) = 2x$ and $\frac{d}{dx}(5) = 0$, this simplifies to: $(x^2 - 4) \cdot 2x = 2x(x^2 - 4)$

Thus, the answer is $2x(x^2 - 4) = 2x^3 - 8x$, which matches option C.

The correct answer is C: \frac{x^3}{3} - 4x.

Would you like more details or have any questions?

Here are 5 related questions:

1. What is the Leibniz rule for differentiating integrals with both variable limits?
2. How do you compute definite integrals involving polynomials?
3. What are the different types of differentiation rules?
4. How does the chain rule apply in Leibniz's rule for variable limits?
5. Can the product rule and chain rule be applied together?

Tip: When solving integrals with variable limits, always consider whether any terms can be simplified or eliminated based on the properties of the limits!