Question

Differentiate \(y=\frac{(x+2)^{2}}{x}\) with respect to \(x\)

Short answer

The derivative is \( \frac{dy}{dx} = 1 - \frac{4}{x^2} \).

Step-by-step solution

Identify the Function Type

The function given is \( y = \frac{(x + 2)^{2}}{x} \). This is a rational function, where the numerator is a polynomial \((x + 2)^{2}\) and the denominator is \(x\).

Simplify the Function, if Possible

We can simplify the function by dividing each term in the numerator by the denominator. This gives us \( y = \frac{x^2 + 4x + 4}{x} \). Simplifying further, \( y = x + 4 + \frac{4}{x} \).

Differentiate Each Term Separately

Differentiate \( y = x + 4 + \frac{4}{x} \) term by term:- The derivative of \( x \) with respect to \( x \) is 1.- The derivative of the constant \( 4 \) is 0.- Rewrite \( \frac{4}{x} \) as \( 4x^{-1} \). The derivative of \( 4x^{-1} \) is \( -4x^{-2} \).

Combine the Derivatives

Combine the derivatives from Step 3:\[ \frac{dy}{dx} = 1 + 0 - 4x^{-2} = 1 - \frac{4}{x^2} \]

Present the Final Derivative

The final derivative of \( y \) with respect to \( x \) is:\[ \frac{dy}{dx} = 1 - \frac{4}{x^2} \]

Key concepts

Rational Functions

Rational functions are an important class of functions in mathematics. They are represented as the ratio of two polynomials. In the function \( y = \frac{(x + 2)^{2}}{x} \), the numerator \((x + 2)^{2}\) is a polynomial, as is the denominator, \(x\). This structure makes it a rational function.Rational functions often exhibit interesting behaviors, such as asymptotes. An asymptote is a line that the graph of the function approaches but never actually touches. In our case, as the denominator approaches zero, the value of the function can grow very large.Understanding rational functions involves:

• Identifying the numerator and denominator polynomials.
• Finding values that might cause the function to be undefined, such as division by zero.

Analyzing these attributes helps in understanding function behavior over different values of \(x\).

Polynomial

A polynomial is a mathematical expression consisting of variables and coefficients, combined using addition, subtraction, and multiplication. They are often given in terms of powers of \(x\). A basic example is \(x^2 + 4x + 4\), which appears in our exercise as the expanded form of the numerator in the rational function.Polynomials have specific properties:

• The degree, which is the highest power of the variable. In \(x^2 + 4x + 4\), the degree is 2.
• They can be simplified or expanded using algebraic operations.

Manipulating polynomials is a crucial skill, enabling us to simplify expressions, as seen when we expanded \((x + 2)^{2}\) to simplify the overall function before differentiation.

Derivatives

The derivative of a function is a fundamental concept in calculus. It represents the rate of change of a function with respect to one of its variables. In this exercise, we differentiate the function \(y = x + 4 + \frac{4}{x}\) with respect to \(x\):

• The derivative \(\frac{dy}{dx}\) of the function measures how \(y\) changes as \(x\) changes.

To differentiate, you need to apply rules like:

• The power rule: \(\frac{d}{dx}[x^n] = nx^{n-1}\).
• The constant rule where the derivative of a constant is zero.

The derivative provides insights into the slope of the tangent line to the curve at any point, which is essential for understanding the function's behavior.

Simplifying Expressions

Simplifying expressions is key to efficiently solving mathematical problems. It involves reducing expressions to their simplest form, making them easier to work with.For rational functions, simplification can involve:

• Expanding polynomials in the numerator.
• Reducing fractions by dividing each term of the numerator by the denominator, if possible.

This exercise showed simplifying \(\frac{(x+2)^2}{x}\) to \(x + 4 + \frac{4}{x}\), making the derivative easier to find.By simplifying, one reduces computational complexity, which in turn helps in applying calculus operations like differentiation more effectively. Simplified forms often provide clearer insights into the function's properties.