Let's solve the problem! The given function is:

$h(x) = \int_1^{\sqrt{x}} \frac{z^2}{z^4 + 1} \, dz.$

We need to use the Fundamental Theorem of Calculus (FTC) to differentiate $h(x)$ with respect to $x$. Here’s how:

Step 1: FTC and differentiation rules

The Fundamental Theorem of Calculus Part 1 states that:

$\frac{d}{dx} \int_a^{g(x)} f(z) \, dz = f(g(x)) \cdot g'(x),$

where $g(x)$ is the upper limit of integration. If the lower limit depends on $x$, the rule becomes:

$\frac{d}{dx} \int_{h(x)}^{g(x)} f(z) \, dz = f(g(x)) \cdot g'(x) - f(h(x)) \cdot h'(x).$

In this case, the lower limit is $1$ (constant), and the upper limit is $\sqrt{x}$.

Step 2: Apply the FTC

Let $f(z) = \frac{z^2}{z^4 + 1}$. Since the lower limit is constant, we only differentiate the upper limit:

$h'(x) = f(\sqrt{x}) \cdot \frac{d}{dx}(\sqrt{x}).$

Step 3: Compute $f(\sqrt{x})$

Substitute $z = \sqrt{x}$ into $f(z) = \frac{z^2}{z^4 + 1}$:

$f(\sqrt{x}) = \frac{(\sqrt{x})^2}{(\sqrt{x})^4 + 1} = \frac{x}{x^2 + 1}.$

Step 4: Differentiate $\sqrt{x}$

$\frac{d}{dx}(\sqrt{x}) = \frac{1}{2\sqrt{x}}.$

Step 5: Combine results

$h'(x) = f(\sqrt{x}) \cdot \frac{1}{2\sqrt{x}}.$

Substitute $f(\sqrt{x}) = \frac{x}{x^2 + 1}$:

$h'(x) = \frac{x}{x^2 + 1} \cdot \frac{1}{2\sqrt{x}} = \frac{x}{2\sqrt{x}(x^2 + 1)}.$

Simplify the numerator $x / \sqrt{x} = \sqrt{x}$:

$h'(x) = \frac{\sqrt{x}}{2(x^2 + 1)}.$

Final Answer:

$h'(x) = \frac{\sqrt{x}}{2(x^2 + 1)}.$

Let me know if you have any questions or need further clarification!

Related Questions:

1. How does the Fundamental Theorem of Calculus apply to definite integrals with variable limits?
2. How would the differentiation change if both limits of the integral depend on $x$?
3. Can you solve $\int_1^{\sqrt{x}} \frac{z^2}{z^4 + 1} \, dz$ explicitly for some specific values of $x$?
4. Why is the chain rule crucial when differentiating functions with composite arguments?
5. What are other examples where FTC is applied in physics or engineering?

Tip:

Always check the dependency of the integration limits on the variable of differentiation when applying FTC!