Problem 54 - University Calculus: Early Transcendentals

Question

Find \(d y / d x\). $$y=\int_{2^{x}}^{1} \sqrt[3]{t} d t$$

Step-by-Step Solution

Answer

The derivative is \(-2^x \ln(2)\)."

Step 1: Identify the Type of Problem

The problem requires us to differentiate a function defined by a definite integral with respect to a variable in its limits of integration. This is an application of the Fundamental Theorem of Calculus and requires the use of Leibniz's Rule for differentiating integrals with variable limits.

Step 2: Apply Leibniz's Rule

Leibniz's rule states that if the limits of a definite integral are functions of a variable, the derivative of the integral with respect to that variable can be found using:\[\frac{d}{dx}\left( \int_{a(x)}^{b(x)} f(t) \, dt \right) = f(b(x)) \cdot b'(x) - f(a(x)) \cdot a'(x)\]For the given problem, where the integral is \(y = \int_{2^x}^{1} \sqrt[3]{t} \, dt\), we identify \(a(x) = 2^x\) and \(b(x) = 1\). The function inside the integral, \(f(t)\), is \(\sqrt[3]{t}\).

Step 3: Differentiate with Respect to x

Substitute \(b(x) = 1\) and \(a(x) = 2^x\) into Leibniz's Rule:\[\frac{dy}{dx} = \sqrt[3]{1} \cdot \frac{d}{dx}[1] - \sqrt[3]{2^x} \cdot \frac{d}{dx}[2^x]\]Calculate the derivatives: \(\frac{d}{dx}[1] = 0\) and \(\frac{d}{dx}[2^x] = 2^x \ln(2)\).

Key Concepts

Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus is a key concept in mathematics that bridges the world of derivatives and integrals.
It allows us to evaluate definite integrals and links the operation of differentiation with integration.
Essentially, it states that if you have a continuous function, you can use its integral to find the antiderivative. This concept is encapsulated in two parts:

• The first part of the theorem tells us that if you take the derivative of an integral of a function, you get the original function back. So, if you have a function f that is continuous on an interval \([a, b]\), the function defined by \( F(x) = \int_{a}^{x} f(t) \, dt \) is an antiderivative of f on \([a, b]\).
• The second part of the theorem provides a way to evaluate a definite integral by using an antiderivative of the function.
  If \( F \) is an antiderivative of \( f \) on the interval \([a, b]\), then \( \int_{a}^{b} f(t) \, dt = F(b) - F(a) \). It beautifully combines differentiation and integration operations.

Understanding this theorem allows us to simplify many problems in calculus where differentiation and integration are involved, highlighting their inverse relationship.

Variable Limits of Integration

When dealing with integrals where the limits are not constants but functions of a variable, these are called integrals with variable limits of integration. This concept extends the power of integration, allowing for more complex and dynamic applications.
With variable limits, the upper and/or lower limits of the integral are functions themselves rather than fixed numbers.

• If you have an integral expressed as \( \int_{a(x)}^{b(x)} f(t) \, dt \), then \( a(x) \) or \( b(x) \) or both are functions of the variable \( x \).
• Variable limits are especially important in applications where the bounds of the quantity you are integrating change, such as in physics for motion over a period of time.

Addressing integrals with variable limits requires special attention when differentiating, as changes in the limits contribute to the derivative of the integral. This is where Leibniz's Rule comes into play, giving a clear method for computing these derivatives.

Differentiation of Integrals

Differentiation of integrals, especially those with variable limits, is an essential technique in calculus that allows us to analyze how integrals change with respect to a variable.
When the limits of an integral depend on the variable you are differentiating with respect to, you cannot apply the Fundamental Theorem of Calculus directly.
Instead, we use Leibniz's Rule, which extends the fundamental theorem by accounting for changing limits.Leibniz’s Rule states the following:

• For an integral with variable limits \( \int_{a(x)}^{b(x)} f(t) \, dt \), the derivative with respect to \( x \) is given by: \[ \frac{d}{dx}\left( \int_{a(x)}^{b(x)} f(t) \, dt \right) = f(b(x)) \cdot b'(x) - f(a(x)) \cdot a'(x) \]
• The above formula shows that the derivative of the integral depends not only on the rate of change of the limits but also on the values of the function at the limits.
• Essentially, this approach allows us to calculate how an integral changes as its limits shift, providing powerful insights in various applications.

The example provided illustrates how to apply this rule step by step to properly differentiate an integral like \( y = \int_{2^x}^{1} \sqrt[3]{t} \, dt \), showcasing practical use.