Problem 47 - Calculus and its applications

Question

Differentiate each function. \(g(t)=\frac{-t^{2}+3 t+5}{t^{2}-2 t+4}\)

Step-by-Step Solution

Answer

The derivative is \(g'(t) = \frac{-t^2 - 18t + 22}{(t^2 - 2t + 4)^2}\).

Step 1: Identify the Rule to Use

Since we need to differentiate a quotient of two functions, we'll use the quotient rule. The quotient rule states that if you have a function of the form \(\frac{u(t)}{v(t)}\), then its derivative is given by \(\frac{u'(t)v(t) - u(t)v'(t)}{(v(t))^2}\).

Step 2: Define \(u(t)\) and \(v(t)\)

Let \(u(t) = -t^2 + 3t + 5\) and \(v(t) = t^2 - 2t + 4\). These are the numerator and denominator of our original function \(g(t)\).

Step 3: Differentiate \(u(t)\)

Find the derivative of \(u(t) = -t^2 + 3t + 5\). The derivative \(u'(t) = -2t + 3\).

Step 4: Differentiate \(v(t)\)

Find the derivative of \(v(t) = t^2 - 2t + 4\). The derivative \(v'(t) = 2t - 2\).

Step 5: Apply the Quotient Rule

Substitute \(u(t)\), \(v(t)\), \(u'(t)\), and \(v'(t)\) into the quotient rule formula. The derivative \(g'(t) = \frac{(-2t + 3)(t^2 - 2t + 4) - (-t^2 + 3t + 5)(2t - 2)}{(t^2 - 2t + 4)^2}\).

Step 6: Simplify the Expression

Simplify the expression by expanding and combining like terms. Expanding gives \((-2t + 3)(t^2 - 2t + 4) = -2t^3 + 4t^2 - 8t + 3t^2 - 6t + 12\) and \((-t^2 + 3t + 5)(2t - 2) = -2t^3 + 2t^2 + 6t^2 - 6t + 10t - 10\). Combine these to simplify the final expression for \(g'(t)\).

Step 7: Final Simplification

After combining like terms, the final derivative \(g'(t) = \frac{-2t^3 + 7t^2 - 14t + 12 - (-2t^3 + 8t^2 + 4t - 10)}{(t^2 - 2t + 4)^2} = \frac{-t^2 - 18t + 22}{(t^2 - 2t + 4)^2}\).

Key Concepts

Differentiation

Differentiation is a fundamental concept in calculus that involves finding how a function changes at any given point. Specifically, we use differentiation to determine the derivative of a function, which indicates the rate of change or the slope of the function. Imagine you're graphing out a function, and you want to know the steepness of the curve at various points—this is what differentiation tells you.

When dealing with differentiation, it's crucial to know the various rules and formulas at your disposal. Basic differentiation involves applying simple rules to functions like powers of a variable, exponential functions, and trigonometric functions.

• For a function like a polynomial, the power rule is often used.
• More complex functions, like those involving division, require rules like the quotient rule.

Understanding which rule to employ can make differentiation a straightforward or complex task, depending largely on how comfortable you are with these rules.

Derivative Calculation

Calculating the derivative is an essential step in understanding a function’s behavior. The derivative of a function gives us the rate at which the function's output is changing concerning its input.

In the exercise, the function given is a quotient, which leads us to use the quotient rule for derivative calculation. This rule is specifically designed to handle the differentiation of a function that has a numerator and a denominator.

• The quotient rule is expressed mathematically as: \[\left( \frac{u}{v} \right)' = \frac{u'v - uv'}{v^2}.\]
• Here, \(u\) and \(v\) are themselves functions of \(t\), and derivatives \(u'\) and \(v'\) refer to the derivatives of \(u(t)\) and \(v(t)\), respectively.

This formula shows us how to take these derivatives systematically. It's crucial to correctly compute these derivatives, as any errors in this part can lead to incorrect results later on.

Step-by-Step Solution

A step-by-step solution guides you through the process of differentiation, especially using complex rules like the quotient rule. This approach helps learners see each transformation and calculation, ensuring clarity and understanding. In our example:

1. **Identify the components:** Recognize the parts of the function involved in the quotient.

• Here, we defined \(u(t)\) as the numerator \(-t^2 + 3t + 5\) and \(v(t)\) as the denominator \(t^2 - 2t + 4\).

2. **Differentiate each component:** Take the derivative of \(u(t)\) and \(v(t)\) separately using standard derivative rules.

• The derivatives are \(u'(t) = -2t + 3\) and \(v'(t) = 2t - 2\).

3. **Apply the quotient rule:** Substitute these derivatives into the quotient rule formula to derive \(g'(t)\).4. **Simplify:** After applying the quotient rule, simplify the expression by expanding, combining like terms, and simplifying further.

• This process leads to the final simplified derivation: \[\frac{-t^2 - 18t + 22}{(t^2 - 2t + 4)^2}\]

Following these steps systematically not only provides you with the correct derivative but also deepens your understanding of how derivatives work, preparing you for more complex functions in the future.