Applying the Quotient Rule to Differentiate a Function: Step-by-Step Example

Quotient Rule: d/dx [ f(x) / g(x) ]

The quotient rule is a formula used to find the derivative of a function that is defined as the ratio of two other functions

The quotient rule is a formula used to find the derivative of a function that is defined as the ratio of two other functions. The derivative of the quotient of two functions can be obtained by following the quotient rule.

The quotient rule states that if f(x) and g(x) are differentiable functions, then the derivative of the quotient f(x) / g(x) with respect to x can be computed as:

d/dx [ f(x) / g(x) ] = ( g(x) * f'(x) – f(x) * g'(x) ) / ( g(x) )^2

Where f'(x) and g'(x) represent the derivatives of f(x) and g(x) with respect to x, respectively.

Let’s break it down step by step:

1. Identify f(x) and g(x) as the numerator and denominator of the function you want to differentiate.

2. Calculate the derivative of f(x) with respect to x, denoted as f'(x).

3. Calculate the derivative of g(x) with respect to x, denoted as g'(x).

4. Apply the quotient rule formula:

d/dx [ f(x) / g(x) ] = ( g(x) * f'(x) – f(x) * g'(x) ) / ( g(x) )^2

5. Simplify the expression if possible.

Here is an example to illustrate how to apply the quotient rule:

Example:
Find the derivative of the function f(x) = x^2 / (2x + 1)

Solution:
1. Identify f(x) and g(x):
In this case, f(x) = x^2, and g(x) = 2x + 1.

2. Calculate the derivative of f(x):
f'(x) = 2x

3. Calculate the derivative of g(x):
g'(x) = 2

4. Apply the quotient rule formula:
d/dx [ f(x) / g(x) ] = ( g(x) * f'(x) – f(x) * g'(x) ) / ( g(x) )^2
= ( ( 2x + 1 ) * (2x) – ( x^2 ) * (2) ) / ( ( 2x + 1 ) )^2

5. Simplify the expression:
d/dx [ f(x) / g(x) ] = ( 4x^2 + 2x – 2x^2 ) / ( 2x + 1 )^2
= ( 2x^2 + 2x ) / ( 2x + 1 )^2

So, the derivative of f(x) = x^2 / (2x + 1) is ( 2x^2 + 2x ) / ( 2x + 1 )^2.

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