Problem 46 - University Calculus: Early Transcendentals

Question

Find \(d y / d x\). $$y=\int_{1}^{x} \frac{1}{t} d t, \quad x>0$$

Step-by-Step Solution

Answer

\( \frac{dy}{dx} = \frac{1}{x} \).

Step 1: Understand the Problem

We need to differentiate the function \( y = \int_{1}^{x} \frac{1}{t} \, dt \) with respect to \( x \). The function given is a definite integral where the variable upper limit is \( x \).

Step 2: Recognize the Use of the Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus tells us that if \( F(t) \) is the antiderivative of \( f(t) \), then \( \frac{d}{dx} \int_a^x f(t) \, dt = f(x) \) given that the integrand \( f \) is continuous at \( x \).

Step 3: Apply the Fundamental Theorem of Calculus

Here, the function inside the integral is \( \frac{1}{t} \). According to the Fundamental Theorem of Calculus, \( \frac{dy}{dx} \) would be \( f(x) \), which means \( \frac{dy}{dx} = \frac{1}{x} \).

Step 4: Write the Final Answer

After applying the Fundamental Theorem of Calculus, we find that the derivative of \( y \) with respect to \( x \) is \( \frac{1}{x} \).

Key Concepts

Derivative

The concept of a derivative is fundamental in calculus. It represents the rate at which a function is changing at any given point. In simple terms, the derivative tells us how fast something is changing.

• For example, if you have a function that describes the position of a car over time, its derivative will tell you the car's speed at any given moment.
• Mathematically, the derivative of a function \( y = f(x) \) with respect to \( x \) is often denoted as \( \frac{dy}{dx} \) or \( f'(x) \).

The process of finding a derivative is called differentiation.
Key to solving problems involving derivatives is understanding how small changes in the input value affect changes in the output value.

In our exercise where \( y = \int_{1}^{x} \frac{1}{t} \, dt \), we are finding \( \frac{dy}{dx} \) through the Fundamental Theorem of Calculus, resulting in \( \frac{1}{x} \). This tells us the rate of change of the integral with respect to \( x \).

Definite Integral

The definite integral of a function provides us a way to calculate the area under a curve between two points, usually over an interval \([a, b]\).

• It is represented as \( \int_{a}^{b} f(t) \, dt \).
• This area interpretation makes definite integrals crucial in applications such as calculating distances or accumulated quantities.

The definite integral accounts for both the magnitude and direction of the area (above or below the x-axis) between the interval.
In our exercise, the definite integral \( \int_{1}^{x} \frac{1}{t} \, dt \) represents the area under the curve \( \frac{1}{t} \) from \( t = 1 \) to \( t = x \). This concept ties directly to applying the Fundamental Theorem of Calculus to get our derivative.

Antiderivative

An antiderivative is a reverse process of differentiation. When we have a function and find its antiderivative, we are essentially determining a function whose derivative is the original function.

• For instance, if \( F'(x) = f(x) \), then \( F(x) \) is an antiderivative of \( f(x) \).
• This is directly related to indefinite integrals, which are denoted \( \int f(x) \, dx \), and give us the family of all possible antiderivatives of \( f(x) \).

The Fundamental Theorem of Calculus connects antiderivatives with definite integrals, allowing one to evaluate definite integrals using antiderivatives.
In the exercise, recognizing the antiderivative of \( \frac{1}{t} \) as \( \ln |t| + C \) (where \( C \) is a constant) helps us use the theorem effectively. However, for the derivative of the definite integral, we use \( \frac{dy}{dx} = \frac{1}{x} \) directly as permitted by the theorem.