Quotient rule and its Associated Formula

It is possible to find the derivative or differentiation of an arbitrary function given as a ratio or division of two differentiable functions using the quotient rule in calculus, as shown in the example below. That is, when we need to compute the derivative of a function of the form: f(x)/g(x), we can use the quotient rule as long as both f(x) and g(x) are differentiable and g(x) is less than zero. In differentiation, the quotient rule is directly related to the product rule and the concept of limits of derivation in differentiation. In the following sections, we will go over the formula for the quotient rule, its proof, and several solved instances to help us grasp it better. Due to a lack of a more precise expression, the quotient rule states that the derivative of a quotient is equal to the ratio of what is obtained by subtracting the numerator times its derivative from the denominator times its derivative to what is obtained by multiplying the square of the denominator times its derivative by the square of the denominator times its derivative. Therefore, if we are presented with an inverse function of the form: f(x) = u(x)/v(x), we may obtain the derivative of this function by using the quotient rule to the function.

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

Definition : 

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 derivative formula for the quotient rule can be used to calculate the derivative or evaluate the differentiation of the quotient of two functions, respectively. The following is the derivative formula for the quotient rule:

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 derivatives of the quotient of two differentiable functions can be found using the quotient formula, which we learnt how to use in the previous section. Consider the quotient rule formula’s proof. The quotient rule formula can be proved in a variety of methods, including,

Using limit and derivative properties

Differentiating implicitly

Applying the chain rule 

Using Derivatives and Limit Properties, Proof of the Formula for the Quotient Rule Use the derivative definition or limits to demonstrate the quotient rule formula. Let f(x) = u(x)/v (x).

f'(x)=limh0[f(x+h)-f(x)]/h

=limh0u(x+h)v(x+h)–u(x)v(x)h

=limh0u(x+h)v(x)-u(x)v(x+h)h.v(x).v(x+h)

=(limh0u(x+h)v(x)-u(x)v(x+h)h.v(x).v(x+h))(limh01v(x).v(x+h))

=v(x)u(x)-u(x)v(x)[v(x)]2

Using Implicit Differentiation to Prove the Quotient Rule Formula 

To prove the quotient rule formula using the implicit differentiation formula, let’s start with a differentiable function f(x) = u(x)/v(x), so u(x) = f(x)v (x). Using the product rule, we get: u'(x) = f'(x)v(x) + f(x)v’ (x). When we figure out f'(x), we get,

f'(x)=u'(x)-f(x)v'(x)v(x)

Substitute f(x),

f'(x)=u'(x)-u(x)v(x)v'(x)v(x)

u'(x)v(x)-u(x)v'(x)[v(x)]2

Chain Rule Proof of the Quotient Rule Formula :

Using the chain rule formula, we can figure out the quotient rule formula. Let f(x) be a function that can be changed so that f(x) = u(x)/v (x).

f(x)=u(x)v-1(x)

using the product rule, f‘(x)=u'(x)v-1(x) +u(x).

(ddx(v-1(x))

Applying the power rule to solve the derivative in the second term, we have,

f’(x)=u'(x)v'(x)–u(x)v(x)v(x)2

f’(x)=ddx[u(x)v(x)]=u'(x)v(x)-u(x)v(x)v(x)2

 Conclusion : 

In calculus, the quotient rule is comparable to the product rule in mathematics, and vice versa. A quotient rule is defined as the ratio of the quantity of the denominator times the derivative of the numerator function minus the quantity of the denominator times the derivative of the numerator function minus the quantity of the denominator When a function’s derivative is divided by the denominator function’s square, the denominator functions’ derivatives are divided by their denominator functions. Since there is no more accurate statement for the rule, the derivative of a quotient is equal to the ratio of the numerator’s subtraction.times its derivative from the denominator times its derivative to what is obtained by multiplying the square of the denominator times its derivative by the square of the denominator times its derivative.