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Magazine Chapter 23: Problem 39
Solve the given problems by finding the appropriate derivatives. Find the derivative of \(y=\frac{x^{2}-1}{x-1}\) by (a) the quotient rule, and (b) by first simplifying the function.
Short Answer
Expert verified
The derivative using both methods is \( y' = 1 \).
Step by step solution
01
Identify the function components for the Quotient Rule
The given function is a fraction, so apply the quotient rule to differentiate it. The quotient rule states that if you have a function \( y = \frac{u}{v} \), the derivative \( y' \) is given by \( y' = \frac{v \cdot u' - u \cdot v'}{v^2} \). Here, identify \( u = x^2 - 1 \) and \( v = x - 1 \).
02
Differentiate the Numerator and Denominator
Find the derivatives of \( u \) and \( v \) with respect to \( x \). For \( u = x^2 - 1 \), \( u' = 2x \). For \( v = x - 1 \), \( v' = 1 \).
03
Apply the Quotient Rule
Substitute \( u, v, u', \) and \( v' \) into the quotient rule formula: \( y' = \frac{(x - 1)(2x) - (x^2 - 1)(1)}{(x - 1)^2} \). Simplify this expression to find \( y' \).
04
Simplify the Quotient Formula
Simplify the expression \( y' = \frac{2x^2 - 2x - x^2 + 1}{(x-1)^2} = \frac{x^2 - 2x + 1}{(x-1)^2} \). Notice that the numerator \( x^2 - 2x + 1 \) is a perfect square, which is \( (x-1)^2 \).
05
Conclusion for Part (a)
The derivative from the quotient rule is: \( y' = \frac{(x-1)^2}{(x-1)^2} = 1 \), for \( x eq 1 \).
06
Simplify the Function for Part (b)
Simplify the original function \( y = \frac{x^2 - 1}{x-1} \) by performing polynomial division. Note that \( x^2 - 1 = (x-1)(x+1) \), thus \( y = x + 1 \) for \( x eq 1 \).
07
Differentiate the Simplified Function
Now that \( y = x + 1 \), find the derivative \( y' \) with respect to \( x \). Since \( y = x + 1 \) is a linear equation, \( y' = 1 \).
08
Conclusion for Part (b)
The derivative of the simplified function is \( y' = 1 \), consistent with the result from part (a).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
quotient rule
Differentiating a function that is presented as a fraction involves using the quotient rule. This rule is a handy technique in calculus to find the derivative of a quotient of two functions. Let's break it down in simple terms.
The quotient rule is represented as follows: If you have a function expressed as \( y = \frac{u}{v} \), the derivative of \( y \) with respect to \( x \) is given by:\[ y' = \frac{v \cdot u' - u \cdot v'}{v^2} \]
The quotient rule is represented as follows: If you have a function expressed as \( y = \frac{u}{v} \), the derivative of \( y \) with respect to \( x \) is given by:\[ y' = \frac{v \cdot u' - u \cdot v'}{v^2} \]
- Identify: Start by identifying the two parts of your fraction鈥攏umerator \( u \) and denominator \( v \).
- Differentiate: Find the derivatives \( u' \) and \( v' \) with respect to \( x \).
- Substitute: Plug these into the quotient rule formula.
- Simplify: Finally, simplify the resulting expression as much as possible.
polynomial division
Polynomial division is an essential skill in simplifying certain mathematical expressions, especially when dealing with rational functions such as \( y = \frac{x^2 - 1}{x-1} \). This method involves breaking down a complex polynomial into simpler components. This process can make differentiation easier by eliminating the need for more complex derivative rules.
Here's a simple approach to polynomial division:
Here's a simple approach to polynomial division:
- Factorize the numerator: Begin by seeing if the numerator can be expressed as a product of its factors. For instance, \( x^2 - 1 = (x-1)(x+1) \).
- Cancel common terms: Simplify the fraction by cancelling common factors in the numerator and the denominator. In this case, \( \frac{(x-1)(x+1)}{x-1} \) simplifies to \( x+1 \), provided \( x eq 1 \).
differentiation by simplifying
Before starting with complex derivative processes, always check if a function can be simplified. Simplification can transform a challenging derivative task into a straightforward one. In the case of rational expressions, consider simplifying the function first to possibly avoid using rules like the quotient rule.
Here's how simplification can be beneficial:
Here's how simplification can be beneficial:
- Transform the Function: By reducing it through methods like polynomial division, you can potentially simplify it to a more basic polynomial or even a linear function.
- Differentiate Easily: Once simplified, derivatives become much easier to manage. For instance, simplifying \( \frac{x^2 - 1}{x-1} \) to \( x + 1 \) allows us to quickly determine that the derivative is simply \( 1 \).
