Chapter 2: Problem 3
Differentiate the following functions: \(y=\frac{x^{2}}{1-x}\)
Short Answer
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Question: Find the derivative of the function \(y = \frac{x^2}{1-x}\).
Answer: The derivative of the function is \(y' = \frac{x(2 - x)}{(1-x)^2}\).
Step by step solution
01
Identify u and v
In this problem, we have \(y = \frac{u}{v}\), where \(u = x^2\) and \(v = 1 - x\).
02
Differentiate u and v
Now we need to differentiate both \(u\) and \(v\).
For \(u = x^2\), we get \(u' = \frac{d}{dx}(x^2) = 2x\).
For \(v = 1 - x\), we get \(v' = \frac{d}{dx}(1 - x) = -1\).
03
Apply the quotient rule
Now we can apply the quotient rule for differentiation: \(y' = \frac{u'v - uv'}{v^2}\).
Substituting \(u = x^2\), \(u' = 2x\), \(v = 1-x\), and \(v' = -1\), we get
\(y' =\frac{(2x)(1-x) - (x^2)(-1)}{(1-x)^2}\)
04
Simplify the expression
Now let's simplify the expression for \(y'\):
\(y' =\frac{2x - 2x^2 + x^2}{(1-x)^2}\)
Combining like terms, we get:
\(y' =\frac{x(2 - x)}{(1-x)^2}\)
So, the first derivative of the given function is:
\(y' =\frac{x(2 - x)}{(1-x)^2}\)
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Quotient Rule
Understanding the quotient rule is essential when differentiating rational functions, which are formed when one polynomial is divided by another. In calculus, the quotient rule is a method to find the derivative of a function that is the quotient of two differentiable functions. Here's how it is generally expressed:
If you have a function in the form of \( y = \frac{u}{v} \), where both \( u \) and \( v \) are functions of \( x \), and both are differentiable, the quotient rule states that:
\[ y' = \frac{u'v - uv'}{v^2} \]
In this formula, \( u' \) and \( v' \) represent the derivatives of \( u \) and \( v \) with respect to \( x \) respectively. Intuitively, the quotient rule tells us to differentiate the numerator and the denominator separately, multiply the top function's derivative with the bottom function, and subtract the product of the bottom function's derivative with the top function. Finally, we'll divide by the square of the bottom function.
If you have a function in the form of \( y = \frac{u}{v} \), where both \( u \) and \( v \) are functions of \( x \), and both are differentiable, the quotient rule states that:
\[ y' = \frac{u'v - uv'}{v^2} \]
In this formula, \( u' \) and \( v' \) represent the derivatives of \( u \) and \( v \) with respect to \( x \) respectively. Intuitively, the quotient rule tells us to differentiate the numerator and the denominator separately, multiply the top function's derivative with the bottom function, and subtract the product of the bottom function's derivative with the top function. Finally, we'll divide by the square of the bottom function.
First Derivative
The first derivative of a function represents the rate at which the function's value changes as its input changes. In other words, it measures the sensitivity of the function's output to small changes in its input. It is often denoted by \( f'(x) \) or \( y' \) when the function is given as \( f(x) \) or \( y \) respectively.
In practice, computing the first derivative involves applying differentiation rules such as the power rule, product rule, and as we've seen, the quotient rule. These rules are tools that help us articulate the behavior of functions in a precise manner. For instance, in the original exercise where we have \( y = \frac{x^2}{1 - x} \), the first derivative \( y' \) signifies how the slope of the tangent line to the curve described by \( y \) changes. Itβs also the key to identifying maximum and minimum points on a graph, which are fundamental to understanding the function's graph.
In practice, computing the first derivative involves applying differentiation rules such as the power rule, product rule, and as we've seen, the quotient rule. These rules are tools that help us articulate the behavior of functions in a precise manner. For instance, in the original exercise where we have \( y = \frac{x^2}{1 - x} \), the first derivative \( y' \) signifies how the slope of the tangent line to the curve described by \( y \) changes. Itβs also the key to identifying maximum and minimum points on a graph, which are fundamental to understanding the function's graph.
Calculus Step by Step
Approaching calculus problems step by step is a hallmark of good mathematical practice. It allows one to break down complex operations into manageable actions and develop a deep understanding of the problem-solving process. When differentiating functions, especially rational functions, following a systematic method ensures consistency and accuracy.
Breaking Down the Steps
- Identify the Functions: Determine what functions comprise the numerator (u) and the denominator (v).
- Differentiate the Functions: Use differentiation rules to find the derivatives of u and v, denoted as \( u' \) and \( v' \).
- Apply the Appropriate Rule: For quotients, use the quotient rule where \( y' = \frac{u'v - uv'}{v^2} \) to find the derivative of the whole function.
- Simplify the Result: Often, you'll need to simplify the expression by combining like terms or factoring.