Problem 9 - University Calculus: Early Transcendentals

Question

Find the indicated derivatives. $$\frac{d s}{d t}\( if \)s=\frac{t}{2 t+1}$$

Step-by-Step Solution

Answer

The derivative is \( \frac{1}{(2t + 1)^2} \).

Step 1: Identify the Type of Function

The given function is a rational function: \( s = \frac{t}{2t+1} \), which consists of a numerator and a denominator.

Step 2: Recall the Quotient Rule

The derivative of a quotient \( \frac{u}{v} \) is found using the quotient rule: \( \frac{d}{dt}\left(\frac{u}{v}\right) = \frac{v\frac{du}{dt} - u\frac{dv}{dt}}{v^2} \). Here, \( u = t \) and \( v = 2t + 1 \).

Step 3: Differentiate the Numerator

Compute the derivative of the numerator \( u = t \). The derivative \( \frac{du}{dt} = 1 \) since the derivative of \( t \) with respect to \( t \) is 1.

Step 4: Differentiate the Denominator

Compute the derivative of the denominator \( v = 2t + 1 \). The derivative \( \frac{dv}{dt} = 2 \) since the derivative of \( 2t \) is 2 and the derivative of a constant 1 is 0.

Step 5: Apply the Quotient Rule

Substitute \( u, v, \frac{du}{dt}, \frac{dv}{dt} \) into the quotient rule formula to calculate \( \frac{ds}{dt} \): \[ \frac{ds}{dt} = \frac{(2t + 1)\cdot 1 - t\cdot 2}{(2t + 1)^2} = \frac{2t + 1 - 2t}{(2t + 1)^2} \].

Step 6: Simplify the Expression

Simplify the expression from Step 5: \( 2t + 1 - 2t = 1 \). So, \( \frac{ds}{dt} = \frac{1}{(2t + 1)^2} \).

Key Concepts

Quotient Rule for Derivatives

When you have a situation where you're dealing with the derivative of a fraction containing two functions—one function in the numerator and another in the denominator—you'll need the quotient rule. It's essential to understand that the quotient rule is like a tool specifically designed for fractions in calculus.

Suppose you have a function which is a fraction \( \frac{u}{v} \), where \( u = t \) and \( v = 2t + 1 \) as in this exercise. The quotient rule states that the derivative \( \frac{d}{dt}\left(\frac{u}{v}\right) \) can be found using:

• Take the derivative of the numerator \( u \) which is \( \frac{du}{dt} \).
• Take the derivative of the denominator \( v \) which is \( \frac{dv}{dt} \).
• Plug these derivatives into the formula: \( \frac{v \cdot \frac{du}{dt} - u \cdot \frac{dv}{dt}}{v^2} \).

Thus, understanding this formula and applying it step-by-step helps us find the derivatives of rational functions effectively.

What Is a Rational Function?

A rational function is simply a fraction made up of two polynomials. In the expression \( s = \frac{t}{2t+1} \), the numerator is \( t \) (a simple polynomial), and the denominator is \( 2t + 1 \) (another polynomial).

Rational functions are quite common in calculus. Their defining feature is this combination of a numerator and denominator, like the fraction you see above. Understanding rational functions is crucial since they behave differently at certain points, especially where the denominator equals zero, potentially causing vertical asymptotes or undefined values.

• Numerator: A polynomial that sits on top of the fraction.
• Denominator: A polynomial on the bottom, determining the division.
• Behavior: Includes finding where the function is undefined when the denominator is zero.

Grasping the structure of rational functions is the first step in successfully applying the quotient rule.

Differentiation Steps for Rational Functions

Differentiation is the process of finding the derivative. For rational functions like \( s = \frac{t}{2t+1} \), these steps are typically involved:

1. **Identify the Components** To use the quotient rule, it's vital first to identify which part of the function is the numerator \( u \) (here \( t \)) and which is the denominator \( v \) (here \( 2t + 1 \)).
2. **Differentiate the Numerator** Compute \( \frac{du}{dt} \). For the example \( u = t \), the derivative is simple: \( \frac{du}{dt} = 1 \).
3. **Differentiate the Denominator** Compute \( \frac{dv}{dt} \). With \( v = 2t + 1 \), its derivative is straightforward too: \( \frac{dv}{dt} = 2 \).
4. **Apply the Quotient Rule** Insert these derivatives into the quotient rule formula as presented: \\[ \frac{ds}{dt} = \frac{(2t + 1) \cdot 1 - t \cdot 2}{(2t + 1)^2} \\] This calculation gives the result for the derivative.
5. **Simplify** Simplification is often necessary for a neat answer. In this case, simplifying gets you \( \frac{1}{(2t + 1)^2} \).
By understanding and following these clear steps, anyone can find the derivatives of rational functions with confidence.