Problem 31 - Calculus Single Variable

Question

In each of Exercises \(29-34,\) calculate the derivative of \(F(x)\) with respect to \(x\) $$ F(x)=\int_{x}^{\pi / 4} \cot (t) d t $$

Step-by-Step Solution

Answer

The derivative of \(F(x)\) is \(-\cot(x)\).

Step 1: Understanding the Problem

The function we are dealing with is an integral of the form \( F(x) = \int_{x}^{\pi/4} \cot(t) \, dt \). We need to find the derivative of this function with respect to \(x\).

Step 2: Applying Leibniz Rule for Differentiation Under the Integral Sign

The Leibniz Rule allows us to differentiate an integral where the limits and possibly the integrand are functions of \(x\). For the integral \( \int_{a(x)}^{b(x)} f(t) \, dt \), the derivative is \( f(b(x))b'(x) - f(a(x))a'(x) \). Here, \( b(x) = \pi/4 \) and \( a(x) = x \).

Step 3: Derivative of Upper Limit

The upper limit \( \pi/4 \) is constant, so its derivative is zero: \( \left(\frac{d}{dx} \pi/4\right) = 0 \).

Step 4: Derivative of Lower Limit

The lower limit is \(x\), so the derivative of \(x\) with respect to \(x\) is 1: \( \left(\frac{d}{dx} x\right) = 1 \).

Step 5: Substituting into the Leibniz Rule

Using the rule, \( f(a(x)) \) becomes \( \cot(x) \) since \( a(x) = x \). Therefore, the derivative \( F'(x) \) results in \(-\cot(x) \times 1 = -\cot(x) \).

Key Concepts

Leibniz Rule

The Leibniz Rule is a powerful tool in differential calculus, particularly useful when dealing with integrals that have limits dependent on a variable, like our function here. It allows us to differentiate an integral without explicitly solving the integral first.

This rule states that if you have an integral of the form \( \int_{a(x)}^{b(x)} f(t) \, dt \), where both the upper \( b(x) \) and lower \( a(x) \) limits are functions of \( x \), the derivative of the integral with respect to \( x \) is given by:

• \( f(b(x))b'(x) \) - representing the part due to the changing upper limit
• - \( f(a(x))a'(x) \) - representing the part due to the changing lower limit

In our case, the function \( a(x) = x \) takes the place of the lower limit and \( f(t) = \cot(t) \) is the integrand. Recognizing which limit affects the integral is crucial in leveraging the Leibniz Rule effectively. By differentiating the limits, we can find the rate of change of this integral, bypassing the need to compute the integral itself.

Derivative of Integrals

The derivative of an integral is a concept that extends the fundamental principles of calculus to integrate and then differentiate functions. This concept can be particularly intriguing when the limits of our integrals are functions themselves, as seen in our example equation.

In simpler cases, with constant limits, the Fundamental Theorem of Calculus implies the derivative of the integral simply recovers the integrand. However, when limits are variable, as with our function \( F(x) = \int_{x}^{\pi/4} \cot(t) \, dt \), Leibniz Rule comes into play.

For the lower limit \( a(x) = x \):

• The derivative with respect to \( x \) returns the negative of the integrand evaluated at this lower limit, \( -\cot(a(x)) \).

This results in \( -\cot(x) \) for our exercise, encapsulating how derivative operations handle the potential changes in integral boundaries gracefully.

Calculus Problem Solving

In calculus problem-solving, particularly when dealing with derivatives of integrals, recognizing the role of each component is key. Our exercise demonstrates this through differentiating the integral \( \int_{x}^{\pi / 4} \cot(t) \, dt \).

The main steps involved include:

• Identifying the function for which the derivative is being sought and recognizing that it is within an integral.
• Applying the Leibniz Rule to account for any changes in the limits caused by the variable \( x \).
• Substituting relevant function values back into formulated expressions to find the derivative.

Problem-solving in calculus requires us to efficiently merge differentiation rules and algebraic manipulations. Understanding these steps solidifies core calculus concepts and aids effectively tackling complex expressions involving derivatives of integrals.