Understanding and Applying the Quotient Rule for Differentiation in Calculus

Quotient Rule

The quotient rule is a useful formula in calculus that allows us to differentiate functions that are written as the quotient of two other functions

The quotient rule is a useful formula in calculus that allows us to differentiate functions that are written as the quotient of two other functions. The rule states that if we have a function f(x) divided by another function g(x), the derivative of this quotient can be found by following a specific formula.

To state the quotient rule, let’s consider two differentiable functions, f(x) and g(x), where g(x) is not equal to zero. The quotient rule states:

\[ \frac{d}{dx} \left(\frac{f(x)}{g(x)}\right) = \frac{g(x) \cdot f'(x) – f(x) \cdot g'(x)}{(g(x))^2} \]

To break down the quotient rule further, we can observe that it consists of three parts:

1. The numerator on the right side of the formula is obtained by taking the derivative of the numerator function f(x) and multiplying it by the denominator function g(x): g(x) * f'(x).

2. The denominator on the right side consists of squaring the denominator function g(x): (g(x))^2.

3. Subtract the product obtained in step 1 from the product of f(x) and g'(x): – f(x) * g'(x).

The quotient rule is useful when finding the derivative of functions that involve division or fractions. By applying this rule, you can determine the rate of change of a function and understand its behavior more precisely.

Let’s illustrate this with an example:

Suppose we want to find the derivative of the function f(x) = (2x^2 + 3x – 1) / (x + 2).

First, we need to identify f(x) and g(x), where f(x) = 2x^2 + 3x – 1 and g(x) = x + 2. We’ll also find their respective derivatives:

f'(x) = 4x + 3
g'(x) = 1

Now, using the quotient rule:

\[ \frac{d}{dx} \left(\frac{f(x)}{g(x)}\right) = \frac{(x+2)(4x+3)-(2x^2+3x-1)(1)}{(x+2)^2} \]

Simplifying further, we get:

\[ \frac{4x^2+11x+6-2x^2-3x+1}{(x+2)^2} = \frac{2x^2+8x+7}{(x+2)^2} \]

This is the derivative of the function f(x) = (2x^2 + 3x – 1) / (x + 2).

So, the quotient rule allows us to find the derivative of a function that involves division or fraction by following a specific formula. By applying this rule, we can calculate the instantaneous rate of change of the function at any given point.

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