Mastering the quotient rule: Step-by-step guide to finding the derivative of a quotient of two functions

d/dx[f(x)/g(x)] (quotient rule)

To find the derivative of the quotient of two functions f(x) and g(x), we can use the quotient rule

To find the derivative of the quotient of two functions f(x) and g(x), we can use the quotient rule. The quotient rule states that the derivative of f(x)/g(x) is given by:

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

So, let’s break down the steps to apply the quotient rule:

Step 1: Identify the functions f(x) and g(x).

Step 2: Differentiate f(x) and g(x) separately. Let’s denote the derivatives as f'(x) and g'(x), respectively.

Step 3: Apply the quotient rule formula: [g(x)*f'(x) – f(x)*g'(x)] / [g(x)]^2

Let’s work through an example to illustrate this process:

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

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

Step 2: Differentiate f(x) and g(x) separately.
f'(x) = 2x (using the power rule for differentiation)
g'(x) = 1 (a derivative of x + 1 is just 1, as it is a linear function)

Step 3: Apply the quotient rule formula: [g(x)*f'(x) – f(x)*g'(x)] / [g(x)]^2
Now, substitute the values derived in the previous steps.

d/dx [x^2 / (x + 1)] = [(x + 1)*(2x) – (x^2)*(1)] / [(x + 1)^2]

Simplifying further, we have:

= (2x^2 + 2x – x^2) / (x^2 + 2x + 1)
= (x^2 + 2x) / (x^2 + 2x + 1)

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

Remember, this is just one example. You can apply the same steps to find the derivative of any function that involves a quotient of two functions.

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