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**Proof of the First Part of the Fundamental Theorem of Calculus**

Let $f$ be a continuous function on the closed interval $[a, b]$ and let $F$ be an antiderivative of $f$ on the interval $(a, b)$. We want to show that $F$ is differentiable on the interval $[a, b]$ and that its derivative is $f$.

Let $x$ be any point in the interval $[a, b]$. We can construct the following Riemann sum:

```
∑_{i=1}^n f(x_i) \Delta x_i
```

where $a = x_0 < x_1 < x_2 < \dots < x_n = b$ and $\Delta x_i = x_{i+1} - x_i$. By the mean value theorem for integrals, we know that there exists a point $c_i \in (x_{i-1}, x_i)$ such that:

```
∑_{i=1}^n f(x_i) \Delta x_i = \int_a^b f(x) \, dx = f(c_i) (b - a)
```

Dividing both sides by $(b - a)$ and taking the limit as $n \to \infty$ and $\Delta x_i \to 0$, we get:


lim_{n \to \infty} \frac{1}{b - a} \sum_{i=1}^n f(x_i) \Delta x_i = \lim_{n \to \infty} f(c_i) = f(x)


But we also know that:

```
lim_{n \to \infty} \frac{1}{b - a} \sum_{i=1}^n f(x_i) \Delta x_i = F'(x)
```

Therefore, we have:

```
F'(x) = f(x)
```

This proves that $F$ is differentiable on the interval $[a, b]$ and that its derivative is $f$.

**Proof of the Second Part of the Fundamental Theorem of Calculus**

Let $f$ be a continuous function on the closed interval $[a, b]$. We want to show that:

```
∫_a^b f(x) \, dx = F(b) - F(a)
```

where $F$ is any antiderivative of $f$ on the interval $(a, b)$.

By the first part of the fundamental theorem of calculus, we know that $F'(x) = f(x)$ for all $x$ in the interval $(a, b)$. This means that:

```
F(x) = ∫_a^x f(t) \, dt
```

for all $x$ in the interval $(a, b)$.

Substituting $x = b$ into the above equation, we get:

```
F(b) = ∫_a^b f(t) \, dt
```

Subtracting $F(a)$ from both sides, we get:

```
F(b) - F(a) = ∫_a^b f(t) \, dt
```

This proves the second part of the fundamental theorem of calculus.

**Conclusion**

We have proven both parts of the fundamental theorem of calculus. This theorem is one of the most important results in calculus, and it has many applications in science and engineering.
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