Mastering the Quotient Rule in Calculus: A Step-by-Step Guide for Finding Derivatives

quotient rule

The quotient rule is a formula used in calculus to find the derivative of a function that is expressed as the quotient of two other functions

The quotient rule is a formula used in calculus to find the derivative of a function that is expressed as the quotient of two other functions. It is derived by applying the chain rule to the division of two functions.

Let’s say we have a function f(x) = g(x) / h(x), where g(x) and h(x) are both differentiable functions.

The quotient rule states that the derivative of f(x), denoted as f'(x), is given by:

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

To derive this formula, we can start by finding the derivative of the numerator and denominator separately using the product rule.

For the numerator, we have d/dx [g(x) * h(x)] = g'(x) * h(x) + g(x) * h'(x)

For the denominator, we have d/dx [h(x)]^2 = 2 * h(x) * h'(x)

Now, we substitute these derivatives into the original formula:

f'(x) = [h(x) * g'(x) + g(x) * h'(x)] / [2 * h(x) * h'(x)]

Next, we can simplify the expression by multiplying both the numerator and denominator by (1/2):

f'(x) = [h(x) * g'(x) + g(x) * h'(x)] / [2 * h(x) * h'(x)] * (1/2)

This gives us:

f'(x) = [h(x) * g'(x) + g(x) * h'(x)] / [2 * h(x) * h'(x)] * (1/2)

Now, we can divide both terms in the numerator by h(x), and cancel out h'(x) in the denominator with h'(x) in the numerator:

f'(x) = [g'(x) + g(x) * h”(x)/h'(x)] / (2 * h(x))

Finally, we can rearrange the terms to get the quotient rule in its standard form:

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

So, this is the formula for the quotient rule. It allows us to find the derivative of a function that is expressed as the quotient of two other functions.

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