Question

Derivatives Find and simplify the derivative of the following functions. $$y=\frac{w^{4}+5 w^{2}+w}{w^{2}}$$

Short answer

Answer: The derivative of the function with respect to w is \(y'(w) = 2w\).

Step-by-step solution

Identify the numerator and denominator functions

Before applying the quotient rule, let's identify the numerator and denominator functions. The given function is: $$y=\frac{w^{4}+5 w^{2}+w}{w^{2}}$$ In this case, the numerator function is \(N(w) = w^{4} + 5w^{2} + w\) and the denominator function is \(D(w) = w^{2}\).

Compute the derivatives of the numerator and denominator functions

Next, we need to compute the derivatives of the numerator and denominator functions with respect to w. The derivative of the numerator function N(w) is: $$N'(w)=\frac{d}{dw}(w^{4}+5w^{2}+w)=4w^3+10w+1$$ The derivative of the denominator function D(w) is: $$D'(w)=\frac{d}{dw}(w^{2})=2w$$

Apply the quotient rule

Now, we will apply the quotient rule to find the derivative of the given function. The quotient rule states that if \(y=\frac{N(w)}{D(w)}\), then \(y'(w)=\frac{N'(w)D(w)-N(w)D'(w)}{[D(w)]^2}\). Applying the quotient rule, we get: $$y'(w)=\frac{(4w^3+10w+1)(w^{2})-(w^{4}+5w^{2}+w)(2w)}{(w^{2})^2}$$

Simplify the expression

Now, we will simplify the y'(w) expression by performing the necessary multiplications and subtractions. $$y'(w)=\frac{(4w^5+10w^3+w^{2})-(2w^5+10w^3+2w^2)}{w^4}$$ Combining the like terms, the derivative simplifies to: $$y'(w)=\frac{2w^5}{w^4}$$ Dividing the power of w, we get: $$y'(w)=2w$$ The simplified derivative of the given function is \(y'(w) = 2w\).

Key concepts

Differentiation

Differentiation is a fundamental concept in calculus, involving the process of finding the derivative of a function. To put it simply, the derivative measures how a function's output value changes as its input value changes. Essentially, it provides the rate at which the function is changing at any given point.

For the given function, the process of differentiation allows us to find the rate of change of the function with respect to its variable, which in this case is w. The derivative describes the slope of the tangent line to the curve at any point, reflecting the function's instantaneous rate of change. Differentiation can be performed using various rules, including the power rule, product rule, quotient rule, and chain rule, depending on the structure of the function being differentiated.

Understanding differentiation is crucial, not just for solving calculus problems but also for its applications across physics, engineering, economics, and other fields where understanding the change is essential.

Simplifying Expressions

Simplifying expressions in calculus is the process of making them as elementary as possible. This often involves reducing fractions, factoring, and cancelling common terms to achieve a more basic and easier-to-understand form.

In the context of derivatives, simplifying the expression after applying rules of differentiation is a crucial step. During simplification, combine like terms, reduce fractions by dividing the numerator by the denominator when they contain common factors, and eliminate any complex expressions.

For the derivative obtained in the given problem, simplification led us from a more complex fraction to the final simplified derivative: \(y'(w) = 2w\). This simplified form is much easier to work with, especially when it comes to further mathematical analysis or application of the derived formulas.

Chain Rule

The chain rule is a powerful technique in differentiation that deals with the derivative of composite functions. When a function is composed of two or more functions, the chain rule allows you to find its derivative by taking the derivative of the outer function and multiplying it by the derivative of the inner function.

In essence, if we have a composite function like \(f(g(x))\), the derivative according to the chain rule would be \(f'(g(x)) \times g'(x)\). While the chain rule wasn't directly used in our original exercise, understanding it is essential for dealing with more complex functions where composition of functions is present.

For students, being able to identify when to use the chain rule and applying it correctly is a necessary skill for efficiently handling a variety of differentiation problems.

Power Rule

The power rule is one of the most straightforward rules in differentiation, used when dealing with functions that involve powers of the variable. According to the power rule, if a function is given by \(f(x) = x^n\), where n is any real number, the derivative of the function is \(f'(x) = n \times x^{n-1}\).

This rule dramatically simplifies the differentiation process for polynomial terms. In the given problem, the power rule was applied to find the derivatives of the numerator and the denominator before using the quotient rule to get the derivative of the whole function. Recognizing when the power rule can be used is a crucial skill, as it simplifies differentiation and avoids the need for more complex rules unless necessary.

Once the derivatives of the numerator and denominator were found, they were applied in the quotient rule to find the derivative of the function as a whole. The power rule is a time-saver and an essential part of a student's calculus toolkit.