Question

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

Short answer

Answer: The derivative of the function is \(f'(w) = 2w - \frac{1}{w^2}\).

Step-by-step solution

Identify u(w) and v(w)

The given function is in the form \(\frac{u(w)}{v(w)}\). Identify the functions \(u(w)\) and \(v(w)\). $$u(w)=w^4+5w^2+w$$ $$v(w)=w^2$$

Find the derivatives u'(w) and v'(w)

Calculate the derivatives of \(u(w)\) and \(v(w)\) with respect to w. $$u'(w)=\frac{d}{dw}(w^4 + 5w^2 + w)=4w^3+10w+1$$ $$v'(w)=\frac{d}{dw}(w^2)=2w$$

Apply the quotient rule

Now, using the quotient rule, find the derivative of the given function: $$\frac{d}{dw}\left(\frac{u(w)}{v(w)}\right)= \frac{u'(w)v(w)-u(w)v'(w)}{[v(w)]^2}$$ Insert the expressions for \(u(w), v(w), u'(w),\) and \(v'(w)\). $$\frac{d}{dw}\left(\frac{w^4+5w^2+w}{w^2}\right) = \frac{(4w^3+10w+1)(w^2)-(w^4+5w^2+w)(2w)}{(w^2)^2}$$

Simplify the expression

Expand the numerator and simplify the expression. Begin by expanding: $$= \frac{4w^5+10w^3+w^2-2w^5-10w^3-2w^2}{w^4}$$ Next, combine like terms: $$= \frac{2w^5 - w^2}{w^4}$$ Simplify the fraction: $$\frac{d}{dw}\left(\frac{w^4+5w^2+w}{w^2}\right) = 2w - \frac{1}{w^2}$$ The derivative of the given function is: $$2w - \frac{1}{w^2}$$

Key concepts

Quotient Rule

The quotient rule is a method in calculus for finding the derivative of a function that is the ratio of two differentiable functions. When you have a function expressed as one part divided by another, like \( f(w) = \frac{u(w)}{v(w)} \), the quotient rule comes into play. To use this rule, we first need to differentiate the numerator, which we call \( u(w) \), and the denominator, referred to as \( v(w) \), independently. The formula for the quotient rule is:
\[ \frac{d}{dw}(f(w)) = \frac{u'(w)v(w)-u(w)v'(w)}{[v(w)]^2} \]
This formula means that the derivative of the function \( f(w) \) is the derivative of the numerator times the denominator, minus the numerator times the derivative of the denominator, all over the square of the denominator. In practice, maintaining the correct order is crucial: it's always 'derivative of the top times the bottom minus the top times derivative of the bottom', never the reverse!

Derivatives of Polynomial Functions

Polynomials are algebraic expressions that consist of terms in the form of \( ax^n \), where \( a \) is a coefficient, \( x \) is a variable, and \( n \) is a non-negative integer, which is the term's degree. Derivatives of polynomial functions are found by applying basic differentiation rules. The power rule is the primary tool for finding derivatives of polynomial functions, which states that if \( f(w) = w^n \), then the derivative \( f'(w) = nw^{n-1} \).
For example, to differentiate \( f(w) = w^4 \), we multiply the exponent by the coefficient (which is 1 in this case) and decrease the exponent by one, giving us \( f'(w) = 4w^3 \). When differentiating polynomials, we apply this rule to each term independently and then sum the results to get the derivative of the whole polynomial.

Simplifying Expressions

Once you've applied the quotient rule and found the derivatives of the numerator and denominator, the next step often involves simplifying expressions to obtain the most uncomplicated form of the derivative. This may include expanding products, combining like terms, and cancelling out common factors. Simplification helps in understanding the behavior of the derivative and in further calculations. For instance, in the above example, after applying the quotient rule, we expanded the terms and combined like terms to simplify the expression. The aim is to end up with a derivative that ideally has no fractions, something that's much easier to handle in subsequent calculus operations or in analyzing the function’s behavior.

Calculus

Calculus is a branch of mathematics focused on limits, functions, derivatives, integrals, and infinite series. It has two main branches: differential calculus and integral calculus, which are related by the fundamental theorem of calculus. Calculus is applied extensively in science, engineering, and economics and is used to solve problems where change is involved. In relation to derivatives, calculus enables us to find rates of change and the slopes of curves on graphs. The process of finding a derivative, such as the problem demonstrated here, is a foundational tool in differential calculus. It allows us to understand how a function changes at any given point and this understanding is critical for solving real-world problems involving optimization and motion.