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

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

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

\(\frac{xe^x - e^x}{x^2}\)

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

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

\(\frac{-2x}{x^4} = \frac{-2}{x^3}\)

\(\frac{2xe^x - x^2e^x}{e^{2x}} = \frac{2x - x^2}{e^x}\)

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

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

The Quotient Rule is a method for finding the derivative of a function that is the ratio of two differentiable functions.

The Quotient Rule applies when differentiating a function of the form f(x)/g(x), where both f(x) and g(x) are differentiable.

A rational function is a function that can be expressed as the quotient of two polynomials.

\(f(x)\) represents the numerator function in the quotient \(\frac{f(x)}{g(x)}\).

\(g(x)\) represents the denominator function in the quotient \(\frac{f(x)}{g(x)}\).

\(f'(x)\) represents the derivative of the function \(f(x)\) with respect to \(x\).

\(g'(x)\) represents the derivative of the function \(g(x)\) with respect to \(x\).

It allows us to find derivatives of complex rational functions that cannot be easily simplified or differentiated using other rules.

While not always required, simplifying the derivative after applying the Quotient Rule is often beneficial for further analysis or calculations.

The Quotient Rule yields the derivative of a rational function, indicating the rate of change of the function.

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