Everything you need to know about the quotient rule!

The Quotient Rule in calculus is a formal rule that determines the derivative (differentiation) of a function by dividing by a differentiable function. It is useful in differentiation problems in which one function is divided by another. In general, this rule is applied at the bottom of the derivative definition. The bottom function is first and then the quadratic is applied. Calculus quotient rules are similar to product rules. 

They represent the ratio of the quantity of the denominator times the denominator function divided by the numerator function divided by the denominator function divided by the square of the denominator function. Using the quotient rule, you can easily differentiate functions or quotients. This is also called quotient rule differentiation in Mathematics. This Rule states that the derivative of a quotient is equal to the denominator times the numerator times the derivative of the denominator plus the numerator times the denominator, all divided by the square of the denominator.

The formula for quotient rule:

Let the given function be f(x), which is given by:

f(x)=a(x)/b(x)’

Thus, the differentiation of the function is given by:

f′(x)=[a(x)/b(x)]′= a(x). b′(x)– a(x).B’(x)/{b(x)}2

The quotient rule of differentiation states that (1st function / 2nd function) is equal to the ratio of (Differentiation of the first function × the 2nd function – Differentiation of the second function × 1st function) to the square of the second function. 

Proof of quotient rule:

In the case where the function f(x) is defined as the product of two functions, say m(x) and n(x), then the derivative can be computed as shown below:

f(x) = m(x)/n(x)

This can also be written as:

f(x) = m(x) [m(x)]-1

Using product rule of differentiation,

f'(x) = m'(x) [n(x)]-1 + m(x) (d/dx) [n(x)]-1

We know that (d/dx)xn = nxn-1,

f'(x) = m'(x) [n(x)]-1 + m(x).(-1)[n(x)]-2 n'(x)

= [m'(x)/n(x)] – [{m(x) n'(x)}/{n(x)}2]

= [m’(x) n(x) – m(x) n'(x)]/ [n(x)]2

This is also called the quotient rule of m and n.

This, the above formula can be written as:

(m/n)’= m’n – mn’/ n 2

How to apply the quotient rule in differentiation? 

We can use the following listed steps to find the derivative of a differentiable function f(x) = u(x)/v(x) using the quotient rule for functions of the form f(x) = u(x)/v(x).

  • First of all, note down the values of u(x) and v(x).
  • Find the values of u’(x) and v’(x) and apply the quotient rule formula, which is expressed as f’(x) = [u(x)/v(x)]’ = [u’(x) × v(x) – u(x) × v’(x)]/[v(x)]2

Example of quotient rule:

The quotient rule can help you find the value of f’(x) for the following function f(x): f(x) = x 2/(x+1).

Here, f(x) = x2/(x + 1)

u(x) = x2

v(x) = (x + 1)

⇒ u'(x) = 2x

⇒ v'(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) = (2×2 + 2x – x2)/(x + 1)2

⇒ f'(x) = (x2 + 2x​​​​)/(x + 1)2

Answer: The derivative of x2/(x + 1) is (x2 + 2x​​​​)/(x + 1)2.

Applicability:

It is possible to use the quotient rule to find the differential or differentiation of an expression in the form of a ratio or a division of two differentiable functions. In other words, if we must find the derivative of a function written as follows: The quotient rule follows directly from the product rule and the concept of limits of derivation in differentiation. f(x)/g(x) ensures that f(x) and g(x) are both differentiable, and g(x) is not equal to 0.It is a way of determining a derivative of a function that is given by the result of dividing two differentiable functions indexed by a quotient. 

This can also be studied through chain rule. Both the methods have been taught very carefully by Cuemath, your ultimate partner for math problems, as solutions are made easy & adjusted according to the learning speed of the student. 

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