Derivative of Quotient of Functions | The Quotient Rule in Calculus

Quotient Rule

The quotient rule is a formula used in calculus to find the derivative of a quotient of two functions

The quotient rule is a formula used in calculus to find the derivative of a quotient of two functions. It is particularly useful when you have a function that can be expressed as the division of two other functions.

The quotient rule states that if you have a function f(x) that can be written as the quotient of two functions, g(x) and h(x), then the derivative of f(x) is given by:

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

In other words, to find the derivative of a quotient of functions, you take the derivative of the numerator, multiply it by the denominator, subtract the product of the numerator and the derivative of the denominator, and then divide the whole expression by the square of the denominator.

Let’s illustrate this with an example:

Suppose we have the function f(x) = (3x^2 + 1) / (2x – 1). To find the derivative using the quotient rule, we need to differentiate both the numerator and the denominator separately.

Let g(x) = 3x^2 + 1 and h(x) = 2x – 1.

Now, we find the derivatives of g(x) and h(x):

g'(x) = 6x
h'(x) = 2

Using the quotient rule, we can find the derivative of f(x):

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

= ((6x * (2x – 1)) – ((3x^2 + 1) * 2)) / (2x – 1)^2

Simplifying further, we can expand and factorize to get the final answer:

= (12x^2 – 6x – 6x^2 – 2) / (2x – 1)^2

= (6x^2 – 6x – 2) / (2x – 1)^2

So, the derivative of f(x) with respect to x is (6x^2 – 6x – 2) / (2x – 1)^2.

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