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How to find the derivative of \(y = \frac{x^2}{x+1}\)?
1. Identify f(x) = x^2 and g(x) = x+1. 2. Find f'(x) = 2x and g'(x) = 1. 3. Apply the Quotient Rule: \(\frac{(x+1)(2x) - (x^2)(1)}{(x+1)^2}\). 4. Simplify: \(\frac{x^2 + 2x}{(x+1)^2}\).
Steps to differentiate \(y = \frac{\sin(x)}{x^2}\)?
1. Identify f(x) = sin(x) and g(x) = x^2. 2. Find f'(x) = cos(x) and g'(x) = 2x. 3. Apply the Quotient Rule: \(\frac{(x^2)(\cos(x)) - (\sin(x))(2x)}{(x^2)^2}\). 4. Simplify: \(\frac{x\cos(x) - 2\sin(x)}{x^3}\).
How to differentiate \(y = \frac{e^x}{\cos(x)}\)?
1. Identify f(x) = e^x and g(x) = cos(x). 2. Find f'(x) = e^x and g'(x) = -sin(x). 3. Apply the Quotient Rule: \(\frac{(\cos(x))(e^x) - (e^x)(-\sin(x))}{(\cos(x))^2}\). 4. Simplify: \(\frac{e^x(\cos(x) + \sin(x))}{\cos^2(x)}\).
Find \(\frac{dy}{dx}\) for \(y = \frac{x}{\ln(x)}\).
1. Identify f(x) = x and g(x) = ln(x). 2. Find f'(x) = 1 and g'(x) = 1/x. 3. Apply the Quotient Rule: \(\frac{(\ln(x))(1) - (x)(1/x)}{(\ln(x))^2}\). 4. Simplify: \(\frac{\ln(x) - 1}{(\ln(x))^2}\).
Steps to find the derivative of \(y = \frac{1}{x^3+1}\)?
1. Identify f(x) = 1 and g(x) = x^3 + 1. 2. Find f'(x) = 0 and g'(x) = 3x^2. 3. Apply the Quotient Rule: \(\frac{(x^3+1)(0) - (1)(3x^2)}{(x^3+1)^2}\). 4. Simplify: \(\frac{-3x^2}{(x^3+1)^2}\).
Differentiate \(y = \frac{x^3 + 2}{x - 1}\).
1. Identify f(x) = x^3 + 2 and g(x) = x - 1. 2. Find f'(x) = 3x^2 and g'(x) = 1. 3. Apply the Quotient Rule: \(\frac{(x-1)(3x^2) - (x^3+2)(1)}{(x-1)^2}\). 4. Simplify: \(\frac{2x^3 - 3x^2 - 2}{(x-1)^2}\).
How to find \(\frac{dy}{dx}\) if \(y = \frac{e^{2x}}{x}\)?
1. Identify f(x) = e^{2x} and g(x) = x. 2. Find f'(x) = 2e^{2x} and g'(x) = 1. 3. Apply the Quotient Rule: \(\frac{(x)(2e^{2x}) - (e^{2x})(1)}{x^2}\). 4. Simplify: \(\frac{e^{2x}(2x - 1)}{x^2}\).
Steps to differentiate \(y = \frac{\cos(x)}{1 + x}\)?
1. Identify f(x) = cos(x) and g(x) = 1 + x. 2. Find f'(x) = -sin(x) and g'(x) = 1. 3. Apply the Quotient Rule: \(\frac{(1+x)(-\sin(x)) - (\cos(x))(1)}{(1+x)^2}\). 4. Simplify: \(\frac{-\sin(x) - x\sin(x) - \cos(x)}{(1+x)^2}\).
Find the derivative of \(y = \frac{x^2 + 1}{x^2 - 1}\).
1. Identify f(x) = x^2 + 1 and g(x) = x^2 - 1. 2. Find f'(x) = 2x and g'(x) = 2x. 3. Apply the Quotient Rule: \(\frac{(x^2-1)(2x) - (x^2+1)(2x)}{(x^2-1)^2}\). 4. Simplify: \(\frac{-4x}{(x^2-1)^2}\).
How to solve: Differentiate \(y = \frac{\tan(x)}{x}\)?
1. Identify f(x) = tan(x) and g(x) = x. 2. Find f'(x) = sec^2(x) and g'(x) = 1. 3. Apply the Quotient Rule: \(\frac{(x)(\sec^2(x)) - (\tan(x))(1)}{x^2}\). 4. Simplify: \(\frac{x\sec^2(x) - \tan(x)}{x^2}\).
Explain the concept of differentiating a quotient.
Differentiating a quotient involves finding the rate of change of a function that is expressed as a ratio of two other functions. The Quotient Rule provides a systematic way to accomplish this.
What is the significance of the denominator squared in the Quotient Rule?
Squaring the denominator accounts for how the rate of change of the denominator affects the overall rate of change of the quotient.
Explain why the order matters in the Quotient Rule formula.
The subtraction in the numerator makes the order important; switching the terms will result in the negative of the correct derivative.
Describe the first step in applying the Quotient Rule.
The first step is to correctly identify the numerator function \(f(x)\) and the denominator function \(g(x)\) in the given quotient.
Describe the second step in applying the Quotient Rule.
The second step is to find the derivatives of both the numerator function \(f'(x)\) and the denominator function \(g'(x)\).
Explain the importance of simplifying the derivative after applying the Quotient Rule.
Simplifying the derivative makes it easier to analyze, interpret, and use in further calculations, such as finding critical points or concavity.
How does the Quotient Rule relate to the Product Rule?
The Quotient Rule can be derived from the Product Rule by rewriting \(\frac{f(x)}{g(x)}\) as \(f(x) \cdot [g(x)]^{-1}\) and applying the Chain Rule.
Explain the 'Low D-High, High D-Low, Draw the Line and Square Below!' mnemonic.
This mnemonic helps remember the order of terms in the Quotient Rule: (Denominator * Derivative of Numerator) - (Numerator * Derivative of Denominator), all divided by (Denominator)^2.
Why is it important to double-check algebra when using the Quotient Rule?
Algebraic errors are common when simplifying the derivative after applying the Quotient Rule, and even a small mistake can lead to an incorrect final answer.
What are some common mistakes when using the Quotient Rule?
Common mistakes include: incorrect order of terms in the numerator, forgetting to square the denominator, and algebraic errors during simplification.
Define the Quotient Rule.
The Quotient Rule is a method for finding the derivative of a function that is the ratio of two differentiable functions.
When does the Quotient Rule apply?
The Quotient Rule applies when differentiating a function of the form f(x)/g(x), where both f(x) and g(x) are differentiable.
What is a rational function?
A rational function is a function that can be expressed as the quotient of two polynomials.
What is \(f(x)\) in the Quotient Rule?
\(f(x)\) represents the numerator function in the quotient \(\frac{f(x)}{g(x)}\).
What is \(g(x)\) in the Quotient Rule?
\(g(x)\) represents the denominator function in the quotient \(\frac{f(x)}{g(x)}\).
What does \(f'(x)\) represent?
\(f'(x)\) represents the derivative of the function \(f(x)\) with respect to \(x\).
What does \(g'(x)\) represent?
\(g'(x)\) represents the derivative of the function \(g(x)\) with respect to \(x\).
Why is the Quotient Rule important?
It allows us to find derivatives of complex rational functions that cannot be easily simplified or differentiated using other rules.
Is simplification after applying the Quotient Rule necessary?
While not always required, simplifying the derivative after applying the Quotient Rule is often beneficial for further analysis or calculations.
What is the result of the Quotient Rule?
The Quotient Rule yields the derivative of a rational function, indicating the rate of change of the function.