Mastering the Quotient Rule: A Step-by-Step Guide to Derivatives of Quotient Functions

Quotient rule of f(x)/g(x)

The quotient rule is a formula used to find the derivative of a function that is expressed as the quotient of two other functions, f(x) and g(x)

The quotient rule is a formula used to find the derivative of a function that is expressed as the quotient of two other functions, f(x) and g(x).

Let’s say we have a function h(x) = f(x) / g(x). The quotient rule states:

h'(x) = (f'(x) * g(x) – f(x) * g'(x)) / (g(x))^2

Here’s a step-by-step breakdown of how to apply the quotient rule:

1. Differentiate the numerator: Compute f'(x), the derivative of the numerator f(x) with respect to x.

2. Differentiate the denominator: Compute g'(x), the derivative of the denominator g(x) with respect to x.

3. Multiply the derivative of the numerator by the denominator: Multiply f'(x) by g(x).

4. Multiply the numerator by the derivative of the denominator: Multiply f(x) by g'(x).

5. Subtract the results from steps 3 and 4: Subtract the product from step 4 from the product from step 3. This gives you the numerator of the derivative.

6. Square the denominator: Take g(x) to the power of 2.

7. Divide the numerator by the denominator: Divide the result from step 5 by the result from step 6. This gives you the derivative of the function h(x).

To summarize, the quotient rule states that the derivative of a function h(x) = f(x) / g(x) is given by h'(x) = (f'(x) * g(x) – f(x) * g'(x)) / (g(x))^2.

Remember to simplify the resulting expression if possible or to factor out common factors.

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