Quotient rule can be defined as the rule of finding out the derivation of two functions when they are in the division.
The formula of quotient rule is given as d / dx [f (x) / g (x)] which is equal to [f’(x) g (x) – f (x) g’(x)] / ( g (x) ) ^2.
Let us discuss the quotient rule with some of the examples.
Consider the first example be: find out the derivative of the given function using the quotient rule. Y = 2 / (x + 1).
The given function is y = 2 / (x + 1). Now we know the formula of quotient rule that is d / dx [f (x) / g (x)] which is equal to [f’(x) g (x) – f (x) g’(x)] / ( g (x) ) ^2. In this function f (x) is 2 and g (x) is (x + 1). Substitute f (x) and g (x) values in the formula then we will get it as d / dx (y) is equal to [(x + 1) d / dx (2) – 2 d / dx (x + 1)] / (x + 1) ^2 if we simplify this then we will get it as dy / dx = [(x + 1) (0) – 2 (1 + 0)] / (x + 1) ^2 which is equal to dy / dx = [ 0 – 2 (1)] / (x + 1) ^2 which is equal to -2 / (x + 1) ^2.
Consider the second example to be: find out the derivative of the given function using quotient rule and the given function is y = x^3 / (3x – 2).
The given function is y = x^3 / (3x – 2) now let us find out the derivative of the given function using quotient rule. We know the formula of quotient rule that is d / dx [f(x) / g(x)] which is equals to [f’(x) g(x) – f(x) g’(x)] / ( g(x) )^2 in this f(x) is x^3 and g(x) is given as (3x – 2) now let us substitute these values in the formula then we will get it as dy / dx = [ (3x – 2) d / dx(x^3) – (x^3) d / dx(3x – 2)] / (3x – 2)^2 if we further simplify this then we will get it as dy / dx = [ (3x – 2) (3x^2) – (x^3) (3(1) – 0) ] / (3x – 2)^2 which is equals to dy / dx = [ (3x – 2) (3x^2) – x^3 (3)] / (3x – 2)^2 in this if we multiply with all the terms then we will get it as dy / dx = [9x^3 – 6x^2 – 3x^3] / (3x – 2)^2 which is equals to dy / dx = [6x^3 – 6x^2] / (3x – 2)^2. This is how we can find the derivative of any fraction by simply using or applying the formula fr he quotient rule. The formula helps in solving many big fractions which involve polynomials or the trigonometric functions. The formula helps in making the steps easier and helps in solving the big mathematicsproblems efficiently.
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