Problem 20 - Calculus - AP Edition

Question

Find \(d x / d t\) $$x=\frac{t^{2}+1}{3 t}$$

Step-by-Step Solution

Answer

The derivative \( dx/dt \) is \( \frac{t^2 - 1}{3t^2} \).

Step 1: Differentiate the numerator

First, differentiate the numerator of the given expression \( t^2 + 1 \) with respect to \( t \). The derivative of \( t^2 \) is \( 2t \) and the derivative of \( 1 \) is \( 0 \), giving us \( 2t \).

Step 2: Differentiate the denominator

Next, differentiate the denominator \( 3t \) with respect to \( t \). The derivative of \( 3t \) is \( 3 \).

Step 3: Apply the Quotient Rule

Use the Quotient Rule for differentiation, which is given by:\[ \frac{d}{dt} \left( \frac{u}{v} \right) = \frac{u'v - uv'}{v^2} \]where \( u = t^2 + 1 \) and \( v = 3t \). We have already found \( u' = 2t \) and \( v' = 3 \).

Step 4: Substitute into Quotient Rule Formula

Substitute \( u = t^2 + 1 \), \( v = 3t \), \( u' = 2t \), and \( v' = 3 \) into the Quotient Rule formula:\[ \frac{d}{dt} \left( \frac{t^2 + 1}{3t} \right) = \frac{(2t)(3t) - (t^2 + 1)(3)}{(3t)^2} \]

Step 5: Simplify the expression

Simplify the expression:\[ \frac{(2t)(3t) - (t^2 + 1)(3)}{(3t)^2} = \frac{6t^2 - 3t^2 - 3}{9t^2} \]

Step 6: Combine and simplify terms

Combine the terms in the numerator:\[ 6t^2 - 3t^2 - 3 = 3t^2 - 3 \]Therefore, the derivative simplifies to:\[ \frac{3(t^2 - 1)}{9t^2} \]

Step 7: Further simplify the expression

Factor the numerator and simplify:\[ \frac{3(t^2 - 1)}{9t^2} = \frac{t^2 - 1}{3t^2} \]Since \(t^2 - 1\) can be written as \((t+1)(t-1)\), further simplification can be done here if necessary. After further simplification, the final result is:\[ \frac{t^2 - 1}{3t^2} \].

Key Concepts

Quotient Rule

The Quotient Rule is a fundamental technique in calculus for differentiating functions that are ratios of two differentiable functions. When you have a function expressed as a fraction \( \frac{u}{v} \), where both \( u \) and \( v \) are functions of \( t \), it's not as straightforward as differentiating \( u \) or \( v \) separately. The Quotient Rule provides a way to do this accurately.

The formula for the Quotient Rule is:

• \[ \frac{d}{dt} \left( \frac{u}{v} \right) = \frac{u'v - uv'}{v^2} \]

This means you:

• Differentiate the numerator \( u \) to get \( u' \).
• Differentiate the denominator \( v \) to get \( v' \).
• Multiply the derivative of the numerator \( u' \) by the original denominator \( v \).
• Subtract the product of the original numerator \( u \) and the derivative of the denominator \( v' \).
• Divide everything by the square of the original denominator \( v^2 \).

Using this rule makes it possible to find the derivative of a quotient without making mistakes in combining terms or simplifying parts incorrectly. It’s particularly useful because it gives you a blueprint that guides you through the differentiating process.

Differentiation Techniques

Differentiation is like finding the "rate of change" of a function and it's a key part of calculus. The main differentiation techniques include using basic rules like the Power Rule, Product Rule, Chain Rule, and Quotient Rule.

- **Power Rule:** If you have a function like \( f(t) = t^n \), you differentiate it to get \( f'(t) = nt^{n-1} \).
- **Product Rule:** Used when you need to differentiate a product of two functions, \( u(t)v(t) \), and is expressed as \( (uv)' = u'v + uv' \).
- **Chain Rule:** Handy when dealing with composite functions like \( f(g(t)) \). It helps you find the derivative by multiplying the derivative of the outer function with the derivative of the inner function.

In the problem at hand, you primarily used the Quotient Rule. Differentiation involves using these techniques as tools. You piece them together depending on the problem structure. Knowing when and how to apply each rule is crucial for efficiently solving calculus problems.

Simplification of Expressions

After applying differentiation techniques, you often end up with a rather complex expression. The final step in many calculus problems is simplifying the derivative you find.

Here's why simplification is important:

• It makes the expression easier to interpret and understand.
• Simplified solutions are easier to use in further calculations or when plugging into other formulas.

For example, once we applied the Quotient Rule to differentiate \( \frac{t^2 + 1}{3t} \), the expression looked complex. Simplification involved combining like terms and reducing fractions where possible. That's how we reduced \( \frac{6t^2 - 3t^2 - 3}{9t^2} \) to \( \frac{3(t^2 - 1)}{9t^2} \).

Finally, recognizing that \( t^2 - 1 \) can be factored into \( (t+1)(t-1) \), led to the most refined form, \( \frac{t^2 - 1}{3t^2} \). Each step eliminated unnecessary complexity and made the solution more elegant.