Quotient rule
Introduction :
The term quotient rule or quotient rule formula is a general term. It is used in calculus (for finding derivatives), in logarithms or in exponents. But when it is used to find the derivative of quotient of two functions it can better be described as derivatives quotient rule or differentiation quotient-rule or Quotient Rule Derivatives. In this article let us discuss that rule relevant to derivatives.
The statement of the rule in calculus in mathematical form is, if f = u/v, then f’ = [u’*v – u*v’]/(v2), where f, u and v are all functions of the same variable. Let us try to give a formal proof. The quotient rule proof can be given by a number of methods.
Since f = u/v, f*v = u. differentiating both sides (with respect to the same common variable) by using product rule, f’*v + f*v’ = u’ or, f’*v + (u/v)*v’ = u’ or, f’*v = u’ – (u/v)*v’= (u’v – u*v’)/(v) and therefore, what follows is, f’= (u’v – u*v’)/(v2).
Another method of proof can be shown based on logarithmic differentiation. If f = u/v, then ln (f) = ln (u/v). By the properties of logarithm, ln (f) = ln (u) –ln (v). Now differentiating both sides (with respect to the same common variable), (1/f)*f’ = (1/u)*u’ – (1/v)*v’ or (1/f)*f’ = (u’/u) – (v’/v) or f’ = (u’/u)*f – (v’/v)*f = (u’/u)*(u/v) – (v’/v)*(u/v) = (u’/v) – (uv’/v2) = (u’v – u*v’)/(v2).
A student must importantly note that the order in the numerator because the numerator is a subtraction operation which is not commutative. Also one must be very careful in operating this rule especially in complicated functions.
For that, many prefer to find alternate ways that may be simpler and easier. For example, if f(x) = (x + 2x2 – 5x3)/(x4), f ‘(x) can be easily found by rewriting the function as f(x) = (x)/(x4) + (2x2)/(x4)– (5x3)/(x4) = x-3 + 2x2 – 5x and differentiate by using power rule, rather than using the quotient rule.
In exponents the rule applicable in case of quotients is very useful. It says, (am)/(an) = a(m –n). The rule can be easily derived by logical reasoning. Because, (am) = product of ‘m’ factors of ‘a’. Similarly, (an) = product of ‘n’ factors of ‘a’.
Therefore in the division (am)/(an), ‘n’ number of factors get cancelled as common factor and what remain is obviously (m – n) factors of ‘a’. In other words, the resultant exponent is (m – n). The same concept is extended to logarithms where the rule for quotients is described as, logb(m/n) = logb(m) – logb(n)
The term quotient rule or quotient rule formula is a general term. It is used in calculus (for finding derivatives), in logarithms or in exponents. But when it is used to find the derivative of quotient of two functions it can better be described as derivatives quotient rule or differentiation quotient-rule or Quotient Rule Derivatives. In this article let us discuss that rule relevant to derivatives.
The statement of the rule in calculus in mathematical form is, if f = u/v, then f’ = [u’*v – u*v’]/(v2), where f, u and v are all functions of the same variable. Let us try to give a formal proof. The quotient rule proof can be given by a number of methods.
Since f = u/v, f*v = u. differentiating both sides (with respect to the same common variable) by using product rule, f’*v + f*v’ = u’ or, f’*v + (u/v)*v’ = u’ or, f’*v = u’ – (u/v)*v’= (u’v – u*v’)/(v) and therefore, what follows is, f’= (u’v – u*v’)/(v2).
Another method of proof can be shown based on logarithmic differentiation. If f = u/v, then ln (f) = ln (u/v). By the properties of logarithm, ln (f) = ln (u) –ln (v). Now differentiating both sides (with respect to the same common variable), (1/f)*f’ = (1/u)*u’ – (1/v)*v’ or (1/f)*f’ = (u’/u) – (v’/v) or f’ = (u’/u)*f – (v’/v)*f = (u’/u)*(u/v) – (v’/v)*(u/v) = (u’/v) – (uv’/v2) = (u’v – u*v’)/(v2).
A student must importantly note that the order in the numerator because the numerator is a subtraction operation which is not commutative. Also one must be very careful in operating this rule especially in complicated functions.
For that, many prefer to find alternate ways that may be simpler and easier. For example, if f(x) = (x + 2x2 – 5x3)/(x4), f ‘(x) can be easily found by rewriting the function as f(x) = (x)/(x4) + (2x2)/(x4)– (5x3)/(x4) = x-3 + 2x2 – 5x and differentiate by using power rule, rather than using the quotient rule.
In exponents the rule applicable in case of quotients is very useful. It says, (am)/(an) = a(m –n). The rule can be easily derived by logical reasoning. Because, (am) = product of ‘m’ factors of ‘a’. Similarly, (an) = product of ‘n’ factors of ‘a’.
Therefore in the division (am)/(an), ‘n’ number of factors get cancelled as common factor and what remain is obviously (m – n) factors of ‘a’. In other words, the resultant exponent is (m – n). The same concept is extended to logarithms where the rule for quotients is described as, logb(m/n) = logb(m) – logb(n)