Quotient Rule Formula

About Quotient Rule Formula

The Quotient rule in calculus is the method used to find the derivative of any function given in form of a quotient obtained from the result, dividing two differentiable functions. We can calculate derivatives or solve the differentiation of the division of two functions using the quotient rule formula. The quotient rule formula is given,

where,

1. f(x) = A function of the form u(x)/v(x) is to be solved.
2. u(x) = A function that makes numerator of the function f(x).
3. u'(x) = Differentiation of u(x).
4. v(x) = A function that makes denominator of the function f(x).
5. v'(x) = Differentiation of the function v(x).

How to Apply Quotient Rule in Derivative?

In order to find the derivative of function of form f(x) = u(x)/v(x). Here, both u(x) and v(x) should be derrivative functions. We can apply following steps to find the differentiation of a derrivative function f(x) = u(x)/v(x) by using quotient rule.

1. Step 1:Note the values of u(x) and v(x).
2. Step 2:Find values of u'(x) and v'(x) and apply the quotient rule formula, given as: f'(x) = [u(x)/v(x)]' = [u'(x) × v(x) - u(x) × v'(x)]/[v(x)]²

Let us have a look at following example given below to understand quotient rule better.

Example: Find f'(x) for the following function f(x) using the quotient rule: f(x) = x²/(x+1).

Solution:

Here,

1. f(x) = x²/(x + 1)
2. u(x) = x²
3. v(x) = (x + 1)
4. ⇒u'(x) = 2x
5. ⇒v'(x) = 1
6. ⇒f'(x) = [v(x)u'(x) - u(x)v'(x)]/[v(x)]²
7. ⇒f'(x) = [(x+1)•2x – x²•1]/(x + 1)²
8. ⇒f'(x) = (2x²+ 2x – x²)/(x + 1)²
9. ⇒f'(x) = (x²+ 2x)/(x + 1)²

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

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