The given function is:

$f(x) = \int_{x^2}^{\sin x} \sqrt{t^2 + 1} \, dt$

This is a definite integral where the limits of integration are functions of $x$, specifically $x^2$ (lower limit) and $\sin x$ (upper limit). To compute the derivative $f'(x)$, we apply the Leibniz rule for differentiation under the integral sign. The rule states:

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

Here:

• $g(t) = \sqrt{t^2 + 1}$,
• $a(x) = x^2$,
• $b(x) = \sin x$.

Step 1: Apply the Leibniz rule

$f'(x) = g(b(x)) \cdot b'(x) - g(a(x)) \cdot a'(x)$

Substitute the terms: $f'(x) = \sqrt{(\sin x)^2 + 1} \cdot (\cos x) - \sqrt{(x^2)^2 + 1} \cdot (2x)$

Step 2: Simplify the result

$f'(x) = \sqrt{\sin^2 x + 1} \cdot \cos x - \sqrt{x^4 + 1} \cdot 2x$

Thus, the derivative of $f(x)$ is: $f'(x) = \cos x \sqrt{\sin^2 x + 1} - 2x \sqrt{x^4 + 1}.$

Let me know if you'd like further details on this process!

Related Questions:

1. Can you compute the second derivative of $f(x)$?
2. How does the Leibniz rule generalize to higher dimensions or more complex integrals?
3. What are some physical applications of definite integrals with variable limits?
4. How does the function $f(x)$ behave as $x$ approaches 0 or infinity?
5. Can we approximate $f(x)$ using Taylor series near a specific point?

Tip:

When differentiating integrals with variable limits, always remember to include the derivatives of the limits in your calculation!