91Ó°ÊÓ

Understanding the derivative of an integral with a variable upper limit is a fundamental concept in calculus. The task is to differentiate a function defined by an integral. In generally speaking, this falls under the purview of the Fundamental Theorem of Calculus.

The theorem suggests that if you have a function defined as an integral, like \( F(x) = \int_{a}^{x} f(t) \, dt \), then its derivative is simply the function inside the integral, evaluated at \( x \). Here, \( F'(x) = f(x) \).

Applying this in our problem: the derivative \( \frac{d}{d x} \int_{2}^{x} \frac{\sin t}{t} \ dt \) becomes \( \frac{\sin x}{x} \). Hence, \( F'(x) \) is the function with respect to \( x \), derived straight from \( \frac{\sin t}{t} \).

When dealing with integrals that have a variable upper limit, you extend the standard integration process. The integral \( \int_{a}^{x} f(t) \ dt \) changes dynamically as \( x \) changes. This process allows you to consider the area under the curve of the function from a fixed point \( a \) to the moving point \( x \).

In our problem, as \( x \) changes, the integral \( \int_{2}^{x} \frac{\sin t}{t} \ dt \) calculates the area starting from constant lower limit \( 2 \) up to a variable point \( x \).

• This concept is essential for understanding how changes in the upper limit affect the value of the integral.
• It illustrates how the derivative of the integral is the function itself at the upper limit.

Solving calculus problems often requires recognizing which theorems and rules to apply. In this exercise, we use the Fundamental Theorem of Calculus to solve for the derivative.

Breaking down this problem:

• Identify the integral expression, which is \( \int_{2}^{x} \frac{\sin t}{t} \ dt \).
• Recognize that you need the derivative of this expression, which translates directly due to the theorem to the function within it, evaluated at \( x \).

A critical aspect of calculus problem solving is

• Decomposing problems into smaller parts.
• Applying known rules or theorems step-by-step.