Problem 41 - Brief Applied Calculus

Question

Find the derivative of each function by using the Quotient Rule. Simplify your answers. $$ f(x)=\frac{x^{2}-2 x+3}{x+1} $$

Step-by-Step Solution

Answer

The derivative of the function is \( f'(x) = \frac{x^2 + 2x - 5}{(x+1)^2} \).

Step 1: Identify Functions in the Quotient Rule

The Quotient Rule for derivatives states: If you have a function \( f(x) = \frac{u(x)}{v(x)} \), then its derivative \( f'(x) \) is given by \( \frac{u'(x)v(x) - u(x)v'(x)}{(v(x))^2} \). Here, \( u(x) = x^2 - 2x + 3 \) and \( v(x) = x + 1 \).

Step 2: Find Derivatives of the Numerator and Denominator

Let's differentiate \( u(x) = x^2 - 2x + 3 \) to find \( u'(x) \) and \( v(x) = x + 1 \) to find \( v'(x) \).- \( u'(x) = 2x - 2 \), because the derivative of \( x^2 \) is \( 2x \), and the derivative of \( -2x \) is \( -2 \).- \( v'(x) = 1 \), because the derivative of \( x \) is \( 1 \).

Step 3: Substitute Into the Quotient Rule Formula

Now, substitute \( u(x) \), \( u'(x) \), \( v(x) \), and \( v'(x) \) into the Quotient Rule formula:\[ f'(x) = \frac{(2x-2)(x+1) - (x^2 - 2x + 3)(1)}{(x+1)^2} \]

Step 4: Simplify the Numerator

Perform the multiplications and subtractions in the numerator:- Multiply \( (2x-2)(x+1) \): \( 2x \cdot x + 2x \cdot 1 - 2 \cdot x - 2 \cdot 1 = 2x^2 + 2x - 2x - 2 = 2x^2 - 2 \)- Subtract the second term: \( (2x^2 - 2) - (x^2 - 2x + 3) = 2x^2 - 2 - x^2 + 2x - 3 \)- Simplify: \( x^2 + 2x - 5 \)

Step 5: Write the Final Derivative

Insert the simplified numerator back into the derivative expression:\[ f'(x) = \frac{x^2 + 2x - 5}{(x+1)^2} \]

Key Concepts

Quotient Rule

The Quotient Rule is a fundamental tool in calculus for finding the derivative of a function that is divided by another function.
It allows you to differentiate expressions that are in the form of a fraction, or more formally, a quotient. When you have a function represented as \( f(x) = \frac{u(x)}{v(x)} \), both \( u(x) \) and \( v(x) \) can be any differentiable functions.
The rule provides a formula for the derivative, which is: \[ f'(x) = \frac{u'(x)v(x) - u(x)v'(x)}{(v(x))^2}\]This formula might look a bit scary at first, but let's break it down:

• \( u'(x) \): the derivative of the numerator function \( u(x) \).
• \( v(x) \) and \( v'(x) \): the denominator function and its derivative.
• The expression \( u'(x)v(x) - u(x)v'(x) \) gives the change in the product, and it is all divided by \((v(x))^2\) to ensure the derivative is relative to the original denominator.

This method is essential for keeping track of how the rates of change of the functions involved interact with each other.

Derivative

A derivative, at its core, is a concept that measures how a function changes as its input changes.
This is often referred to as the "rate of change" or the "slope" at a particular point on the graph of a function. When dealing with a polynomial function like \( u(x) = x^2 - 2x + 3 \), finding the derivative involves applying basic principles, such as the power rule:

• The derivative of \( x^n \) is \( nx^{ (n-1) } \).
• For constants, the derivative is zero.

For our function \( u(x) \), using these rules results in \( u'(x) = 2x - 2 \) because:

• The derivative of \( x^2 \) is \( 2x \).
• The derivative of \(-2x\) is \(-2\).
• The constant \(3\) disappears.

For \( v(x) = x + 1 \), the derivative is \( v'(x) = 1 \) since it's simply a linear function.
Understanding derivatives allows you to predict how changes in input affect output, making it an invaluable tool in calculus.

Simplification

Simplification is the process of making expressions easier to work with by reducing complexity.
When dealing with derivatives, especially those involving the quotient rule, simplification helps render them into more manageable and comprehensible forms.In the provided exercise, after applying the quotient rule, the numerator becomes more extensive and requires simplifying:

• Firstly, expand and combine like terms from operations such as multiplication and subtraction.
• For example, expanding \((2x-2)(x+1)\) results in \(2x^2 - 2\).
• Combining terms \((x^2 + 2x - 5)\) involves careful arithmetic and algebraic manipulation to ensure every sign and term is correct.

Ultimately, simplification aids in writing the derivative in its simplest form, \( \frac{x^2 + 2x - 5}{(x+1)^2} \), making it much more straightforward to interpret and use.
This final form is more elegant and functional for practical applications or further analysis.