In calculus, the quotient rule is a technique for determining the derivative of any function provided in the form of a quotient derived by dividing two differentiable functions. The quotient rule says that the derivative of a quotient is equal to the ratio of the result achieved by subtracting the numerator times the denominator’s derivative from the denominator times the numerator’s derivative to the square of the denominator’s derivative.
Explain quotient rule formula:
Using the quotient rule derivative formula, we can calculate the derivative or evaluate the differentiation of two functions. The derivative formula for the quotient rule is:
f'(x) = [u(x)/v(x)]’ = [v(x) × u'(x) – u(x) × v'(x)]/[v(x)]2
Quotient Rule Formula Derivation
We learned how to use the quotient formula to obtain the derivatives of the quotient of two differentiable functions in the previous section. Let’s have a look at the proof of the quotient rule formula. There are several ways to prove the quotient rule formula, including,
Using limit and derivative properties
Taking use of implicit differentiation
Making use of the chain rule
Properties:
– The quotient rule formula follows the chain rule.
– The quotient rule formula is the reciprocal of the product rule, which is stated as d/dx(u*v) = u d/dx v + u d/dy v
Explain implicit differentiation to prove quotient rule formula:
Using limit properties to prove this result would not be helpful because there are no limits involved in calculating derivatives via implicit differentiation. However, if we accept that if y’ = f(x,y), then dy/dx = y’/f'(x,y), then dy/dx can be calculated using derivatives by taking use of the chain rule. Using this technique will allow us to calculate dy dx without having to use limits. Let u=f(x,y) and v=g(x,y). We know that dy/dx = y’ so we can take the derivative of both sides to get dy dx.
dy dx = y’ dy dx
Uses:
– By using the quotient rule, we can calculate the derivative of some complicated functions that cannot be evaluated easily.
– In some cases, quotient rule is used as an alternative to product rule.
Where:
u(x) = a derivative of a 2nd function
v(x) = a derivative of a 1st function
u'(x) = u'(a)
v'(x) = v'(a)
f'(a) = f ‘(c)’ is equal to one of the following derivatives: d/dx (ax+b), d/dy (ax+b), or du dx +dv dy. This depends on whether there are more variables in v(x) or u'(x).
A real-life example of quotient rule:
When the derivative is 2x+4, we can take it apart like this: (2)(x) + (1)(-4)
2x = -6
The quotient rule can be applied to this function with u(x)=2 and v(x)=-4. Therefore, du/dx=-8 and dv/dx=12. We apply these results to get: -8(-2)+12(-1). We then just have to add this up to get our final result of -20. This may seem a little lengthy but it only takes a few minutes once you’ve done a few of them by hand first.
In baseball statistics, there exists a statistic called a player’s batting average. Batting average is the number of hits divided by the number of at-bats, or BA = H / AB.
H = a derivative of a 2nd function
AB = a derivative of 1st function
H / AB can be rewritten as A(x) differentiation with respect to x . This proves that the quotient rule does, in fact, work. In this case, u(x)= A(x), v(x)=1 and u'(x)=-A'(x). Therefore, du/dx= -(-A’)+(1)(A’)=A’. We apply the result to get our final result being A'(H / AB). We can now say that batting average is equal to a derivative of a 2nd function.If one is facing any problem regarding any mathematical concept all he has to do is just open the Cuemath website and his all-mathematical problem can be solved in no time. Cuemath is like fun math classes online.
