We learn how to take a derivative by using the quotient rule. Using the proper ratio or division of two differentiable functions, the quotient rule in calculus allows one to compute the derivative or differentiation of a process.
Thus, we can apply the quotient rule when searching for derivatives of functions in form f(x)/g(x), such that g(x) * x and f(x) are differentiable.
Directly following the product rule and the concept of limits of derivation in differentiation, the quotient rule follows. Solve examples to explain the formula quotient rule; let’s examine it in detail in the following sections.
For any two functions f(x) and g(x), the derivative of the quotient f(x) / g(x) is given by: (f'(x)g(x) – f(x)g'(x)) / g^2(x)
The derivatives of f(x) and g(x) to x are represented by f'(x) and g'(x), respectively.
It represents the function g(x) raised to a power of two in mathematics. The sum of g(x) and itself is equal to g(x). A table can be plotted as follows: if g(x) = 2x, then g^2(x) = 2x * 2x = 4x^2.
It is important to note that the notation g^2(x) must not be confused with g(x^2), this latter is the composition of g(x) and x^2.
How does the Quotient Rule work?
In calculus, the quotient rule is a method for finding a function’s derivative when given the result of dividing two differentiable functions by their quotient.
Quotient rule states that derivatives are equal to subtracting the numerator from the denominator, then adding the square of the numerator to the denominator.
The quotient rule then allows us to find the derivative of f(x) = u(x)/v(x) if we the function of this form.
The Quotient Rule Formula is derived as follows:
How to find the derivatives of the quotient of two differentiable functions using the quotient formula.
Here is the proof of this formula. The quotient rule formula can provide in different ways, including:
- Limit and derivative properties
- Differentiation implicit
- Chain rule
When finding the derivative of a two-function quotient, we use quotient rules.
It follows that quotient f(x)/g(x) from differencing f and g is also differentiable. It writes as (gf′ * fg′)/g2 in an abbreviated notation.
A function’s derivative is equal to the square of the denominator minus the square of the numerator minus the square of the denominator divided by the square of the denominator. Read Also What is the Vertical Angles
What is the Quotient Rule Derivative?
In order to find the derivative of a function of the form f(x) = u(x)/v(x), u(x) and v(x) must be differentiable functions.
The quotient rule is a method for finding the derivative function as a quotient or a fraction.
The quotient rule states that for any functions f(x) and g(x), where g(x) is not equal to 0, the derivative of the quotient f(x) / g(x) is given by:
(f'(x)g(x) – f(x)g'(x)) / g^2(x)
Let’s break down this expression:
f'(x)g(x) represents the product of the derivative of f(x) and g(x), while f(x)g'(x) represents the product of f(x) and the derivative of g(x).
The numerator of the final expression (f'(x)g(x) – f(x)g'(x)) represents the difference between these two products, which is known as the term “limiting difference.”
And the denominator g^2(x) is the function g(x) squared.
So this formula gives us a way to find the derivative of the quotient of two functions.
It’s a way to relate the slope of the function f(x)/g(x) to the slopes of f(x) and g(x) and their behavior at the point you are trying to find the derivative.
By applying the following steps, we can solve for f(x), where f(x) is a differentiable function defined by u(x)/v(x).
- In order to begin, note down the values of u(x) and v(x).
- The quotient rule formula is given as: f'(x) = [u(x)/v(x)]’ = [u'(x) – u(x) – v'(x)]/[v(x)]2
- To better understand the quotient rule, let’s look at the following example.