How to use FTC1 to find the derivative of a function defined as an integral?

The Fundamental Theorem of Calculus (FTC) Part 1 (FTC1) is a powerful tool that connects differentiation and integration. It states that if a function, say $F(x)$, is defined as the integral of another function $f(x)$ from a constant $a$ to $x$, then the derivative of $F(x)$ is $f(x)$. In other words, if $F(x) = \int_a^x f(t) \, dt$, then $F'(x) = f(x)$.

Here's a step-by-step guide on how to use FTC1 to find the derivative of a function defined as an integral:

1. Identify the integral: Start with the given function $F(x)$ which is defined as an integral. For example, $F(x) = \int_0^x \sin(t) \, dt$.

2. Apply FTC1: According to FTC1, the derivative of $F(x)$ with respect to $x$ is equal to the integrand evaluated at $x$. In other words, $F'(x) = f(x)$.

3. Find the integrand: The integrand is the function inside the integral. In our example, the integrand is $\sin(t)$.

4. Evaluate the integrand at $x$ : Now, evaluate the integrand at $x$. This gives us $\sin(x)$.

So, the derivative of $F(x)$ using FTC1 is $F'(x) = \sin(x)$.