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: $\frac{x^2 + 2x}{(x+1)^2}$.

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: $\frac{x\cos(x) - 2\sin(x)}{x^3}$.

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: $\frac{e^x(\cos(x) + \sin(x))}{\cos^2(x)}$.

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: $\frac{\ln(x) - 1}{(\ln(x))^2}$.

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: $\frac{-3x^2}{(x^3+1)^2}$.

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: $\frac{2x^3 - 3x^2 - 2}{(x-1)^2}$.

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: $\frac{e^{2x}(2x - 1)}{x^2}$.

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: $\frac{-\sin(x) - x\sin(x) - \cos(x)}{(1+x)^2}$.

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: $\frac{-4x}{(x^2-1)^2}$.

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: $\frac{x\sec^2(x) - \tan(x)}{x^2}$.

$\frac{d}{dx} \Bigg[\frac{f(x)}{g(x)}\Bigg] = \frac{g(x)f'(x) - f(x)g'(x)}{(g(x))^2}$

$\frac{d}{dx} \Bigg[\frac{u}{v}\Bigg] = \frac{vu' - uv'}{v^2}$

$\frac{dy}{dx} = \frac{g(x)f'(x) - f(x)g'(x)}{(g(x))^2}$

$\frac{xe^x - e^x}{x^2}$

$\frac{x\cos(x) - \sin(x)}{x^2}$

$\frac{\cos(x) + x\sin(x)}{\cos^2(x)}$

$\frac{-2x}{x^4} = \frac{-2}{x^3}$

$\frac{2xe^x - x^2e^x}{e^{2x}} = \frac{2x - x^2}{e^x}$

$\frac{1 - \ln(x)}{x^2}$

$\frac{(x-1) - (x+1)}{(x-1)^2} = \frac{-2}{(x-1)^2}$

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.

Squaring the denominator accounts for how the rate of change of the denominator affects the overall rate of change of the quotient.

The subtraction in the numerator makes the order important; switching the terms will result in the negative of the correct derivative.

The first step is to correctly identify the numerator function $f(x)$ and the denominator function $g(x)$ in the given quotient.

The second step is to find the derivatives of both the numerator function $f'(x)$ and the denominator function $g'(x)$.

Simplifying the derivative makes it easier to analyze, interpret, and use in further calculations, such as finding critical points or concavity.

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.

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.

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.

Common mistakes include: incorrect order of terms in the numerator, forgetting to square the denominator, and algebraic errors during simplification.