Problem 39 - Brief Applied Calculus

Question

Find the derivative of each function by using the Quotient Rule. Simplify your answers. $$ f(s)=\frac{s^{3}-1}{s+1} $$

Step-by-Step Solution

Answer

The derivative of \( f(s) = \frac{s^3 - 1}{s + 1} \) is \( \frac{2s^3 + 3s^2 + 1}{(s + 1)^2} \).

Step 1: Identify the functions

In the given function \( f(s) = \frac{s^3 - 1}{s + 1} \), identify the numerator and the denominator. Here, \( u(s) = s^3 - 1 \) and \( v(s) = s + 1 \). These will be used for applying the Quotient Rule.

Step 2: Differentiate the numerator and the denominator

Find the derivative of the functions identified in the previous step. The derivative of the numerator, \( u(s) = s^3 - 1 \), is \( u'(s) = 3s^2 \). The derivative of the denominator, \( v(s) = s + 1 \), is \( v'(s) = 1 \).

Step 3: Apply the Quotient Rule

The Quotient Rule states that \( \left(\frac{u}{v}\right)' = \frac{u'v - uv'}{v^2} \). Substitute the derivatives and functions into this rule: \[ \left(\frac{s^3 - 1}{s + 1}\right)' = \frac{(3s^2)(s + 1) - (s^3 - 1)(1)}{(s + 1)^2} \].

Step 4: Simplify the expression

Expand and simplify the expression from Step 3: \[ (3s^2)(s + 1) - (s^3 - 1) = 3s^3 + 3s^2 - s^3 + 1 \]. Combine like terms to simplify: \[ 2s^3 + 3s^2 + 1 \]. So the derivative is \[ \frac{2s^3 + 3s^2 + 1}{(s + 1)^2} \].

Key Concepts

Derivative

The derivative of a function tells us how the function is changing at any given point. It's like finding the slope or the rate of change of the function. Imagine you're riding up a hill on a bicycle. The steeper the hill, the harder you have to pedal. Similarly, the derivative gives us a measure of how steep the graph of a function is at any point. For functions expressed as fractions, like \[ f(s)=\frac{s^{3}-1}{s+1} \] understanding derivatives means using special rules to find out how changes in the variable affect the function's output. Tackling such problems means breaking down the components into easy-to-handle steps.

Simplifying Expressions

Simplifying expressions in the context of derivatives helps us to get a cleaner and more understandable form of the derivative. After calculating the derivative using something like the Quotient Rule, you're often left with a complex expression. It's important to simplify it to avoid mistakes and to make the derivative more usable.Once you find the unsimplified form \[ \frac{(3s^2)(s + 1) - (s^3 - 1)(1)}{(s + 1)^2} \]use algebraic rules to simplify it. Expand the terms in the numerator:

• Multiply 3s^2 by both s and 1, giving 3s^3 + 3s^2.
• Subtract s^3 - 1 from this result to get 2s^3 + 3s^2 + 1.

By simplifying this expression, you get the final derivative:\[ \frac{2s^3 + 3s^2 + 1}{(s + 1)^2} \]which is much easier to interpret and use.

Differentiation Techniques

The Quotient Rule is a crucial technique for differentiation, especially for functions that are fractions. In math terms, when you have a function \[ f(x) = \frac{u(x)}{v(x)} \]where both the top and bottom are differentiable, you can use the formula:\[ \left(\frac{u}{v}\right)' = \frac{u'v - uv'}{v^2} \]This rule helps you find the derivative of the quotient of two functions. For instance, in our example, defining:

• \( u(s) = s^3 - 1 \) and \( u'(s) = 3s^2 \)
• \( v(s) = s + 1 \) and \( v'(s) = 1 \)

The rule systematically handles both functions, meaning you don't mix up their derivatives as you organize them into the formula. So, the Quotient Rule is your go-to method in calculus for tackling fractioned functions effectively.