Problem 39 - Calculus

Question

Evaluate by any method. $$ \frac{d}{d x} \int_{x}^{1} \frac{d t}{t} $$

Step-by-Step Solution

Answer

The derivative is \(-\frac{1}{x}\).

Step 1: Understand the Problem

We need to differentiate the integral \( \int_{x}^{1} \frac{dt}{t} \). This involves finding \( \frac{d}{dx} \int_{x}^{1} \frac{dt}{t} \). Since the upper limit is a constant and the lower limit is a variable, we can use Leibniz rule for differentiation of parameterized integrals.

Step 2: Apply Leibniz Rule

The Leibniz rule states that if we have an integral \( \int_{a(x)}^{b(x)} f(t) \, dt \), then the derivative with respect to \( x \) is \( f(b(x)) b'(x) - f(a(x)) a'(x) \). In our case, the integral is \( \int_{x}^{1} \frac{dt}{t} \) with \( a(x) = x \) and \( b(x) = 1 \). Thus, we have \( \frac{1}{b(x)} \cdot 0 - \left(\frac{1}{a(x)} \cdot 1\right) = -\frac{1}{x} \).

Step 3: Simplify the Expression

From the application of the Leibniz Rule, the result simplifies to \( -\frac{1}{x} \). There is no further simplification necessary as this is the derivative of the given integral with respect to \( x \).

Key Concepts

Leibniz Rule

The Leibniz Rule is a powerful tool in calculus that helps us differentiate an integral when one or both of its limits are functions of a variable. It's especially handy when dealing with parameterized integrals.
To apply the Leibniz Rule to an integral of the form \( \int_{a(x)}^{b(x)} f(t) \, dt \), the rule provides a method for finding the derivative with respect to \( x \).

• The formula derived from the rule is: \( f(b(x)) b'(x) - f(a(x)) a'(x) \).
• This means you evaluate the function \( f(t) \) at the upper limit \( b(x) \) and multiply by the derivative of \( b(x) \), then subtract the evaluation at the lower limit \( a(x) \) multiplied by the derivative of \( a(x) \).

In simpler terms, the Leibniz Rule allows you to handle integrals dynamically as their boundaries change with respect to a variable. It's essential in dealing with real-world systems where boundaries are not static. In our specific example of the function \( \int_{x}^{1} \frac{d t}{t} \), the application of the Leibniz Rule yields the result \( -\frac{1}{x} \). The upper limit, being a constant, contributes nothing to the differentiation, simplifying the calculations significantly.

Parameterized Integrals

Parameterized integrals are integrals where one or both of the limits are not constants, but instead depend on a variable. This creates a more dynamic scenario for integration since the area being integrated beneath can shift as the parameter changes.
These types of integrals are crucial in understanding systems where the conditions change over time or space, making them an essential part of advanced calculus and applied mathematics.

• In parameterized integrals, evaluating the area under a curve requires careful consideration of each limit's dependency on a variable.
• The derivative of such integrals involves applying rules that account for these changing limits, such as the Leibniz Rule.

When determining the derivative with respect to a variable, these integrals consider how changing limits alter the function's overall behavior. In the problem \( \int_{x}^{1} \frac{d t}{t} \), the lower limit changes with \( x \), leading to the application of techniques designed for functions with moving boundaries. Understanding parameterized integrals is a stepping stone to mastering more complex calculus concepts.

Derivative with Respect to a Variable

When we talk about taking the derivative with respect to a variable, we are looking to understand how a function changes as we change one of its inputs. In cases where the function is an integral with variable limits, these derivatives help track these changes in a more complex system.
Given a function integrated over a parameter, differentiating helps pinpoint how the collective output varies as the boundaries of integration move.

• This process unveils the rate of change of the integral with respect to the parameter, providing insights into dynamic changes in areas under curves.
• It is pivotal in understanding how functions grow, shrink, or transform based on modifying inputs, especially when modeled by integrals with variable limits.

In the example \( \int_{x}^{1} \frac{dt}{t} \), differentiating with respect to \( x \) using the Leibniz Rule shows precisely how the area, or accumulation, responds to tweaks in \( x \). The result \( -\frac{1}{x} \) uncovers the integral's sensitivity to shifts in the lower limit, giving us detailed knowledge of its behavior under changing conditions.