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compute the derivative (with respect to x) of the following function


The function given is:

\[
F(x) = \int_{0}^{\tan x} \frac{dt}{1 + t^2}.
\]

To compute the derivative \( F'(x) \) with respect to \( x \), we can use the **Fundamental Theorem of Calculus** combined with the **Chain Rule**.

### Step 1: Fundamental Theorem of Calculus
The Fundamental Theorem of Calculus states that if \( F(x) = \int_{a(x)}^{b(x)} f(t) \, dt \), then:

\[
F'(x) = f(b(x)) \cdot b'(x) - f(a(x)) \cdot a'(x).
\]

In this case, \( f(t) = \frac{1}{1 + t^2} \), \( a(x) = 0 \), and \( b(x) = \tan x \).

### Step 2: Applying the Theorem
Since \( a(x) = 0 \), \( a'(x) = 0 \), so the term involving \( a(x) \) will disappear. We only need to consider the upper limit \( b(x) = \tan x \).

\[
F'(x) = \frac{1}{1 + (\tan x)^2} \cdot \frac{d}{dx}(\tan x).
\]

### Step 3: Simplification
The derivative of \( \tan x \) with respect to \( x \) is \( \sec^2 x \). Therefore:

\[
F'(x) = \frac{1}{1 + \tan^2 x} \cdot \sec^2 x.
\]

Now, recall the trigonometric identity:

\[
1 + \tan^2 x = \sec^2 x.
\]

This simplifies the expression:

\[
F'(x) = \frac{\sec^2 x}{\sec^2 x} = 1.
\]

### Final Answer
The derivative of \( F(x) \) with respect to \( x \) is:

\[
F'(x) = 1.
\]

Would you like to explore any more details or have any further questions?

Here are some related questions to consider:
1. What if the upper limit was \( \sin x \) instead of \( \tan x \)?
2. How would you differentiate \( F(x) = \int_{x}^{\tan x} \frac{dt}{1 + t^2} \)?
3. What is the derivative of \( \int_{a}^{b(x)} f(t) \, dt \) with respect to \( x \) when \( a \) is a constant?
4. How does the derivative change if the integrand is more complex, like \( \frac{dt}{1 + t^4} \)?
5. What are some applications of the Fundamental Theorem of Calculus in physics?

**Tip:** The Fundamental Theorem of Calculus is powerful for evaluating integrals and their derivatives, especially when the limits of integration depend on the variable of differentiation.

Learn how to compute the derivative of the function F(x) = Integral from 0 to tan x of 1 / (1 + t^2) using the Fundamental Theorem of Calculus and the Chain Rule. Step-by-step solution included. Suitable for advanced high school students and calculus enthusiasts.

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