To solve the problem of computing the derivative of the given integral with respect to $x$, we apply the Leibniz rule for differentiation under the integral sign. The given expression is:

$\frac{d}{dx} \left( \int_{5x}^{x^2} \cos(t^2) \, dt \right)$

Step 1: Apply the Leibniz Rule

The Leibniz rule for the derivative of an integral with variable limits is:

$\frac{d}{dx} \left( \int_{a(x)}^{b(x)} f(t) \, dt \right) = f(b(x)) \cdot b'(x) - f(a(x)) \cdot a'(x)$

Here:

• $f(t) = \cos(t^2)$
• The lower limit $a(x) = 5x$ and $a'(x) = 5$
• The upper limit $b(x) = x^2$ and $b'(x) = 2x$

Step 2: Apply the Rule to the Given Problem

Using the Leibniz rule, we get:

$\frac{d}{dx} \left( \int_{5x}^{x^2} \cos(t^2) \, dt \right) = \cos((x^2)^2) \cdot 2x - \cos((5x)^2) \cdot 5$

Step 3: Simplify the Expression

$\cos((x^2)^2) = \cos(x^4)$ $\cos((5x)^2) = \cos(25x^2)$

Thus, the derivative simplifies to:

$2x \cos(x^4) - 5 \cos(25x^2)$

Final Answer:

$\frac{d}{dx} \left( \int_{5x}^{x^2} \cos(t^2) \, dt \right) = 2x \cos(x^4) - 5 \cos(25x^2)$

Would you like further details on any step, or do you have any questions?

Here are 5 related questions to expand on this topic:

1. How does the Leibniz rule change if the limits of the integral are constants?
2. What happens if the function inside the integral is more complex, such as a product of functions?
3. Can you explain the significance of the Fundamental Theorem of Calculus in relation to this problem?
4. How would the derivative change if one of the limits was a trigonometric function of $x$?
5. What are some common applications of using the Leibniz rule in physics or engineering?

Tip: Always double-check the expressions for the limits when applying the Leibniz rule, as small errors in these details can lead to incorrect results.