Problem 20 - Thomas Calculus

Question

Find the derivatives of the functions. $$ f(t)=\frac{t^{2}-1}{t^{2}+t-2} $$

Step-by-Step Solution

Answer

The derivative is \( f'(t) = \frac{t^2 - 2t + 1}{(t^2 + t - 2)^2} \).

Step 1: Identify the Rule

The function can be expressed as a quotient of two functions, where the numerator is \( t^2 - 1 \) and the denominator is \( t^2 + t - 2 \). We need to use the quotient rule for differentiation.

Step 2: Apply Quotient Rule

The quotient rule states that the derivative of a function \( \frac{u}{v} \) is \( \frac{u'v - uv'}{v^2} \). Here, \( u = t^2 - 1 \) and \( v = t^2 + t - 2 \). We need to find \( u' \) and \( v' \).

Step 3: Differentiate the Numerator

Calculate the derivative of the numerator \( u = t^2 - 1 \). The derivative \( u' \) is \( 2t \), since the derivative of \( t^2 \) is \( 2t \) and the derivative of a constant \(-1\) is 0.

Step 4: Differentiate the Denominator

Calculate the derivative of the denominator \( v = t^2 + t - 2 \). The derivative \( v' \) is \( 2t + 1 \), where the derivative of \( t^2 \) is \( 2t \), the derivative of \( t \) is 1, and the derivative of a constant \(-2\) is 0.

Step 5: Substitute into the Quotient Rule Formula

Substitute the derivatives \( u' \) and \( v' \) back into the quotient rule: \[f'(t) = \frac{(2t)(t^2 + t - 2) - (t^2 - 1)(2t + 1)}{(t^2 + t - 2)^2}\]

Step 6: Simplify the Expression

Simplify the expression for \( f'(t) \):First, expand the terms:\[(2t)(t^2 + t - 2) = 2t^3 + 2t^2 - 4t\]and\[(t^2 - 1)(2t + 1) = 2t^3 + t^2 - 2t - 1\]Subtract the expanded result:\[2t^3 + 2t^2 - 4t - (2t^3 + t^2 - 2t - 1) = t^2 - 2t + 1\]Finally, the derivative is:\[ f'(t) = \frac{t^2 - 2t + 1}{(t^2 + t - 2)^2} \]

Key Concepts

Quotient Rule

When you have a function that is the division of two separate algebraic expressions, the perfect tool for differentiation is the quotient rule. The quotient rule is a method in calculus used to find the derivative of a ratio, or quotient, of two functions, say \( \frac{u}{v} \). The derivative is found by applying the following formula: \( \frac{u'v - uv'}{v^2} \), where:

• \( u \) is the numerator of the function, and \( u' \) is its derivative.
• \( v \) is the denominator of the function, and \( v' \) is its derivative.

This rule is incredibly useful when dealing with fractions because it directly computes their derivative without transforming the fraction into another function form.
Using the quotient rule involves identifying the two parts of the function you want to differentiate. For the example given \( f(t) = \frac{t^2 - 1}{t^2 + t - 2} \), \( u = t^2 - 1 \) and \( v = t^2 + t - 2 \). The next steps include differentiating both \( u \) and \( v \), and then substituting them back into the quotient rule formula. It's a systematic approach, especially for more complex expressions.

Differentiation

Differentiation is a core concept in calculus and a method to determine how a function changes at any given point. It's like recognizing the rate of change of the function or how steep the curve is at any segment. Understanding the rules of differentiation helps compute derivatives systematically.
In our function, \( f(t) = \frac{t^2 - 1}{t^2 + t - 2} \), you need to differentiate both the numerator \( t^2 - 1 \) and the denominator \( t^2 + t - 2 \). This involves applying basic differentiation rules:

• The derivative of \( t^2 \) is \( 2t \) using the power rule, which says that \( \frac{d}{dt} (t^n) = n \cdot t^{n-1} \).
• The derivative of a constant \, −1 \, is 0 because constants don't change, hence they contribute nothing to the rate of change.

Using these insights:

• The derivative of the numerator \( u' \) is \( 2t \).
• The derivative of the denominator \( v' \) is \( 2t + 1 \).

These derivatives are then plugged into the quotient rule formula to proceed with finding the derivative of the entire function. Differentiation reveals those critical points where the function behaves differently, making this technique essential for analyzing functions.

Simplifying Expressions

After applying the differentiation rules and methods like the quotient rule, the resulting function can look rather complex. Simplifying expressions is the next crucial step to make sense of the derivative you calculate. Simplification involves reducing fractions, combining like terms, and eliminating any unnecessary parts.
From our example, after applying the quotient rule, the derivative \( f'(t) \) was initially described by a complex fraction. You are required to expand terms in both the numerator and denominator:

• For the numerator: expand and simplify \((2t)(t^2 + t - 2) - (t^2 - 1)(2t + 1)\). This involves distributing and combining like terms, which results in \( t^2 - 2t + 1 \).
• For the denominator: it remains as \( (t^2 + t - 2)^2 \), but ensure no further simplification can happen.

Presenting the derivative in a less complicated form makes it easier to interpret and further analyze, whether you're finding critical points or investigating the behavior of the function graphically. Simplification helps in revealing the more straightforward representation of the function's derivative, which is easier to use in subsequent calculations.