Mastering the Quotient Rule: How to Find the Derivative of a Quotient in Calculus

Quotient Rule

The quotient rule is a formula used for finding the derivative of a quotient of two functions

The quotient rule is a formula used for finding the derivative of a quotient of two functions. It is useful when you have a function that is divided by another function, and you want to find the instantaneous rate of change of that function.

The quotient rule can be stated as follows:

If f(x) and g(x) are functions of x, and g(x) is not equal to zero, then the derivative of f(x)/g(x) is given by the formula:

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

In simpler terms, the derivative of a quotient is obtained by taking the derivative of the numerator, multiplying it by the denominator, subtracting the product of the original numerator and the derivative of the denominator, and then dividing the whole expression by the square of the denominator.

Let’s take an example to better understand how to apply the quotient rule. Consider the function h(x) = (2x^2 + 3x – 4)/(x^2 + 1). To find its derivative, we can use the quotient rule:

First, let’s find the derivative of the numerator, which is f'(x):

f'(x) = d/dx (2x^2 + 3x – 4)
= 4x + 3

Next, let’s find the derivative of the denominator, which is g'(x):

g'(x) = d/dx (x^2 + 1)
= 2x

Now, we can apply the quotient rule formula:

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

= ((4x + 3) * (x^2 + 1) – (2x^2 + 3x – 4) * (2x)) / (x^2 + 1)^2

Simplifying this expression further will give us the derivative function h'(x) for our given example.

So, to sum up, the quotient rule is a formula used to find the derivative of a quotient of two functions. It involves taking the derivative of the numerator, multiplied by the denominator, subtracting the product of the original numerator and the derivative of the denominator, and then dividing the whole expression by the square of the denominator.

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