Quotient Rule in Differentiation

One way to obtain an expression’s derivative or differentiation in calculus is to take the ratio or division of two differentiable functions and divide it by the expression. By this, I mean that while trying to get the derivative of f(x)/g(x), we can use the rule of quotients, because both of these functions are differentiable and g(x) is equal to zero. Products and derivation limitations in differentiation are closely linked to the rule’s application. In the following sections, we’ll go over the formula for the quotient rule and its proof using solved instances.

Apply Quotient Rule in Differentiation

For the function f(x) = u(x)/v(x) to have a derivative, both u(x) and v(x) must be differentiable functions. Using the quotient rule, we can use the steps below to find the derivation of a function f(x) = u(x)/v(x) that can be changed.

Step 1: Write down what u(x) and v are (x).

Second, Find the values of u'(x) and v'(x), and then use the quotient rule formula: f'(x) = [u(x)/v(x)]. ‘ = [u(x) – u(x) – v(x)] /[v(x)]²

Let’s look at the example below to get a better idea of how the quotient rule works.

Quotient Rule

In Calculus, the Quotient rule is comparable to the product rule in mathematics, and vice versa. Specifically, a Quotient Rule is defined as the product of the quantity of the denominator times the derivative of the numerator function divided by the quantity of the denominator times the derivative of the numerator function. Numerator function derivative less the sum of both the denominator and denominator function derivatives. To put it another way, the quotient rule is a way of telling apart quotients from the division of functions. In mathematics, the process of separating two quotient rules is referred to as quotient rule differentiation or simply quotient rule differentiation.

Quotient Rule Formula

The quotient rule derivative formula can be used to compute the derivative or evaluate the differentiation of a quotient of two functions. The formula for the quotient rule derivative is as follows:

f(‘x) = [u(x)/v(x)]’ = [ v(x) × u'(x) – u(x) × v'(x)] / [v(x)]² 

where,

f(x) = The function whose derivative is to be determined, of the form u(x)/v(x).

u(x) = A differentiable function that is the numerator of f. (x).

u'(x) = Function u’s derivative (x).

v(x) = A differentiable function that makes the provided function f’s denominator (x).

v'(x) = Function v’s derivative (x).

Derivation of Quotient Rule Formula

The quotient formula was discussed in depth in the previous chapter. This formula was used to get functions that can be divided by a differentiable quotient which is what we learned about in this section. In the following section, we will look at how to display the quotient rule formula. There are a variety of methods for demonstrating the quotient rule formula, including, but not limited to, the following examples:

Using limit and derivative properties

Differentiating implicitly

Applying the chain rule

Examples of Quotient Rule

Use the quotient rule to find f'(x) for the following function f(x): f(x) = x²/(x+1).

Solution:

Here,  f(x) = x²/(x +1)

u(x) = x²

v(x) = (x +1) 

=> f'(x) = [ v(x)u'(x) – u(x)v’ (x) ] / [ v(x) ]²

=> f'(x) = [ (x+1).2x – x². 1]/ (x+1)²

=> f'(x) = ( 2x² + 2x – x²) / (x+1)²

=> f'(x) = (x² + 2x) / (x+1)²

Apply Quotient Rule in Differentiation 

For the function f(x) = u(x)/v(x) to have a derivative, both u(x) and v(x) must be differentiable functions. Using the quotient rule, we can use the steps below to find the derivation of a function f(x) = u(x)/v(x) that can be changed.

Step 1: Write down what u(x) and v are (x).

Step 2: Find the values of u'(x) and v'(x), and then use the formula for the quotient rule, which is: u(x)/v(x) = f'(x). ‘ = [u(x) – u(x) – v(x)] /[v(x)]²

Conclusion

One way to obtain an expression’s derivative or differentiation in calculus is to take the ratio or division of two differentiable functions and divide it by the expression. By this, I mean that while trying to get the derivative of f(x)/g(x), we can use the rule of quotients, because both of these functions are differentiable and g(x) is equal to zero. In Calculus, the Quotient rule is comparable to the product rule in mathematics, and vice versa. Specifically, a Quotient Rule is defined as the product of the quantity of the denominator times the derivative of the numerator function divided by the quantity of the denominator times the derivative of the numerator function. Numerator function derivative less the sum of both the denominator and denominator function derivatives. The quotient formula was discussed in depth in the previous chapter. This formula was used to get functions that can be divided by a differentiable quotient which is what we learned about in this section. In the following section, we will look at how to display the quotient rule formula.