Problem 25 - Applied Calculus

Question

Find the derivative. Assume that \(a, b, c\), and \(k\) are constants. $$ z=\frac{1-t}{1+t} $$

Step-by-Step Solution

Answer

The derivative of the function is \( \frac{-2}{(1+t)^2} \).

Step 1: Identify the Type of Problem

The problem is to find the derivative of the function \( z = \frac{1-t}{1+t} \). It is a rational function which means we'll likely need to use the quotient rule for differentiation.

Step 2: Recall the Quotient Rule

The quotient rule states that if you have a function \( h(t) = \frac{u(t)}{v(t)} \), then its derivative is \( h'(t) = \frac{u'(t)v(t) - u(t)v'(t)}{(v(t))^2} \), where \( u(t) \) and \( v(t) \) are the numerator and denominator functions respectively.

Step 3: Assign Functions to Numerator and Denominator

For \( z = \frac{1-t}{1+t} \), assign \( u(t) = 1-t \) and \( v(t) = 1+t \).

Step 4: Differentiate the Numerator and Denominator

Find the derivatives of both functions. For \( u(t) = 1-t \), the derivative \( u'(t) = -1 \). For \( v(t) = 1+t \), the derivative \( v'(t) = 1 \).

Step 5: Apply the Quotient Rule

Plug the derivatives into the quotient rule: \[ z'(t) = \frac{(-1)(1+t) - (1-t)(1)}{(1+t)^2} \].

Step 6: Simplify the Expression

Simplify the expression: \( z'(t) = \frac{-1-t-(1-t)}{(1+t)^2} = \frac{-2}{(1+t)^2} \).

Step 7: Final Result Verification

Ensure each step logically follows and that the simplification is correct. The derivative of \( z \) with respect to \( t \) is \( z'(t) = \frac{-2}{(1+t)^2} \).

Key Concepts

Quotient Rule

In the world of calculus, the quotient rule is a powerful tool for finding the derivatives of rational functions, where one function is divided by another. Imagine you have a function expressed as the ratio of two differentiable functions. This is where the quotient rule comes into play, enabling you to tackle the differentiation process effectively.

To visualize it, if you have a function like \( h(t) = \frac{u(t)}{v(t)} \), its derivative is given by the quotient rule —

• \( h'(t) = \frac{u'(t)v(t) - u(t)v'(t)}{(v(t))^2} \)

Here, \( u(t) \) and \( v(t) \) are the two functions in the numerator and denominator, respectively. The rule helps in pinpointing the rate of change of complex functions where the straightforward application of derivative rules might not suffice. It's crucial to practice simplifying outcomes after applying the quotient rule to ensure your answers are both neat and comprehensible.

Rational Functions

Rational functions are mathematical expressions involving the division of two polynomials. They are represented in the form \( \frac{N(t)}{D(t)} \), where \( N(t) \) is the numerator and \( D(t) \) is the denominator, both of which are polynomials. Understanding how to manipulate these expressions is key to mastering the application of derivatives to them.

These types of functions commonly appear in calculus due to their interesting properties and behaviors in different domains. For instance, they can have asymptotes and discontinuities, making them intriguing candidates for differentiation. When differentiating a rational function, techniques like the quotient rule become indispensable.

• The quotient rule allows us to handle the complex interplay between the numerator and denominator.
• It is particularly useful since individual derivatives of numerators and denominators can be easier to compute.

Ultimately, improving your understanding of these functions can greatly aid in solving calculus problems more fluidly.

Differentiation Techniques

Differentiation techniques are essential tools that allow us to compute the derivative of a function, which represents the rate at which the function's value changes. Among these techniques, the quotient rule is specifically tailored for rational functions. It allows us to decompose the problem of taking derivatives into manageable chunks.

Beyond the quotient rule, there are several other key differentiation techniques, each serving unique purposes:

• The **product rule** is used when differentiating products of two functions.
• The **chain rule** is necessary for composite functions where one function is nested within another.
• The **power rule** simplifies finding derivatives of polynomial functions.

These rules often interconnect in complex problems, requiring a solid understanding of each to effectively and efficiently derive solutions. Practicing these techniques will deepen your calculus toolset, helping you to approach a variety of differentiation problems with confidence.