Problem 37 - College Algebra and Calculus: An Applied Approach

Question

Find the derivative of the function. State which differentiation rule(s) you used to find the derivative. $$ g(s)=\frac{s^{2}-2 s+5}{\sqrt{s}} $$

Step-by-Step Solution

Answer

The derivative of the function \(g(s) = \frac{s^{2}-2s+5}{\sqrt{s}}\) is \( g'(s) = \frac{2s^{\frac{3}{2}} + s - 2\sqrt{s} - \frac{s^{2}}{2\sqrt{s}} - \frac{5}{2\sqrt{s}}}{s}\). The Quotient Rule for differentiation was applied to find this derivative.

Step 1: Identify the Top Function and the Bottom Function

The top function of \(g(s)\) is \(f(s) = s^{2} - 2s + 5\) and the bottom function is \(h(s) = \sqrt{s}\). It's crucial to identify these two functions as it will simplify the process of finding the derivative.

Step 2: Apply The Quotient Rule

The quotient rule states that the derivative of \(\frac{f}{h}\) is \(\frac{f'h - fh'}{(h)^{2}}\). Here, the derivative \(f'\) is \(2s - 2\) and the derivative \(h'\) is \(\frac{1}{2\sqrt{s}}\). Plug these values in to get: \(\frac{(2s - 2)\sqrt{s} - (s^{2} - 2s + 5)(\frac{1}{2\sqrt{s}})}{s}\)

Step 3: Simplify The Equation

Simplify the equation to get the final answer. \(g'(s) = \frac{2s^{\frac{3}{2}} - 2\sqrt{s} - \frac{s^{2}}{2\sqrt{s}} + s - \frac{5}{2\sqrt{s}}}{s}\)

Step 4: Further Simplify

After further simplifying, the derivative is given by \( g'(s) = \frac{2s^{\frac{3}{2}} + s - 2\sqrt{s} - \frac{s^{2}}{2\sqrt{s}} - \frac{5}{2\sqrt{s}}}{s}\). The derivative cannot be further simplified because of the existence of \(\sqrt{s}\) in the denominator of some terms.

Key Concepts

Quotient Rule

When dealing with a division of two functions, such as in the problem you have, we use a technique in derivative calculus known as the "quotient rule." The quotient rule helps us find the derivative of a quotient of two functions. It's expressed as follows:

• If you have a function expressed as \( g(s) = \frac{f(s)}{h(s)} \), where \( f(s) \) and \( h(s) \) are differentiable functions, the derivative \( g'(s) \) is computed using the formula: \[ g'(s) = \frac{f'(s)h(s) - f(s)h'(s)}{(h(s))^2} \]

This formula requires you to know both \( f'(s) \), the derivative of the numerator, and \( h'(s) \), the derivative of the denominator.
Applying this rule doesn't have to be complicated: you just need to carefully manage the algebra involved, making sure you subtract the two terms in the numerator correctly and divide by the square of the denominator function.

Simplifying Derivatives

Derivatives can often look a bit messy, especially after applying the quotient rule. Therefore, simplifying derivatives is crucial. This involves combining like terms, reducing fractions, and making the equation look cleaner.
In this exercise, after applying the quotient rule, you'll end up with a long expression containing multiple terms, with some terms having square roots in them. A good first step in simplifying this derivative is looking for ways to combine similar terms.
Sometimes, it may not be possible to simplify so neatly, especially if radicals are involved, but any attempt to reduce and simplify is beneficial, especially for interpretation and further use of the derivative.

Function Differentiation

Function differentiation refers to the process of finding the derivative of a function. It's about understanding how a function's output changes as its input changes.
For the exercise provided, it's all about using existing rules and knowledge to differentiate each component. First, identify your functions \( f(s) = s^2 - 2s + 5 \) and \( h(s) = \sqrt{s} \). The derivatives you'll need are \( f'(s) = 2s - 2 \) and \( h'(s) = \frac{1}{2\sqrt{s}} \).
It's crucial to calculate these correctly before applying the quotient rule. Each function has its own nuances and understanding their derivatives separately helps manage the overall computation.

Algebraic Simplification

Once you have the derivative expression, you're left with algebraic simplification. This is the part where you take a complex equation from differentiation and try to bring it to its simplest form.
This specific task often involves distributing, adding, subtracting, and reducing terms. You will likely encounter fractions and radicals that need breaking down or combining.

• For example, after using the quotient rule, combining like terms such as \( 2s^{\frac{3}{2}} \) and \( -2\sqrt{s} \) is essential.
• It may involve multiplying through by common denominators to simplify any fractional terms further.

In some circumstances, due to complexity, the expression may not simplify tidily but doing as much as possible can significantly ease understanding and utilization.