Chapter 33: Problem 11
Differentiate \(y=\frac{(x+2)^{2}}{x}\) with respect to \(x\)
Short Answer
Expert verified
The derivative is \( \frac{dy}{dx} = 1 - \frac{4}{x^2} \).
Step by step solution
01
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 \(x\).
02
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, \( y = x + 4 + \frac{4}{x} \).
03
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 \( -4x^{-2} \).
04
Combine the Derivatives
Combine the derivatives from Step 3:\[ \frac{dy}{dx} = 1 + 0 - 4x^{-2} = 1 - \frac{4}{x^2} \]
05
Present the Final Derivative
The final derivative of \( y \) with respect to \( x \) is:\[ \frac{dy}{dx} = 1 - \frac{4}{x^2} \]
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
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.
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.
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.
- The power rule: \(\frac{d}{dx}[x^n] = nx^{n-1}\).
- The constant rule where the derivative of a constant is zero.
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.