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To find the derivative or differentiation of a function given as a ratio or division of two differentiable functions, one can use the calculus quotient rule. Because both f(x) and g(x) are differentiable, we can use the formula f(x) g(x) for the derivative of a function of the form f(x)/x, where g(x) > 0. After the product rule and differentiation’s concept of limits on derivation, the quotient rule follows directly on its heels. This section explains the formula for quotient rule and its proof using solved examples.When two differentiable functions are divided, the quotient rule in calculus is used to find the derivative of the smaller of the two functions. Using words, we can say that if we subtract the numerator from the denominator and then multiply the numerator’s derivative by the denominator, then the derivative of the numerator is equal to a quotient’s derivative. A function of this form is called f(x), and the derivative of this function can be computed using the formula f(u(x)/v(x) for the quotient rule.
f'(x) = [u(x)/v(x)]’ = [v(x) × u'(x) – u(x) × v'(x)]/[v(x)]2
Quotient Rule Formula
The quotient rule derivative formula can be used to compute the derivative or evaluate the differentiation of a quotient of two functions. The formula for the quotient rule derivative is as follows:
f'(x) = [u(x)/v(x)]’ = [v(x) × u'(x) – u(x) × v'(x)]/[v(x)]2
where,
f(x) = The function whose derivative is to be determined, of the form u(x)/v(x).
u(x) = A differentiable function that is the numerator of f. (x).
u'(x) = Function u’s derivative (x).
v(x) = A differentiable function that makes the provided function f’s denominator (x).
v'(x) = Function v’s derivative (x).
Derivation of Quotient Rule Formula
The quotient formula, which was discussed in depth in the previous chapter. This formula was used to get functions that can be divided by a differentiable quotient which is what we learned about in this section. In the following section, we will look at how to display the quotient rule formula. There are a variety of methods for demonstrating the quotient rule formula, including, but not limited to, the following examples:
Using limit and derivative properties
Differentiating implicitly
Applying the chain rule
Finding the quotient of two functions
•Step 1: Divide one function by the other to get the quotient of the two functions.
(f/g)(x) = f(x)/g(x)
•Step 2: Identify values that are not in the quotient’s domain. Any values that result in division by zero, for example, cannot be substituted into the quotient.
•Step 3: To fully simplify the quotient, cancel any like factors. Step 2 values that were not in the domain are still not in the domain.
Example: finding the quotient of two functions
1.find (f/g)(x) and state the restrictions on the domain if
f(x) = (x+4)(x-7) and g(x) = 2(x +4)
Step 1: Divide one function by the other to get the quotient of the two functions.
(f/g)(x) = f(x)/g(x)
By dividing the first function by the second, we may obtain the quotient of the two functions.
(f/g)(x) = [(x+4)(x-7)]/2(x+4)
Step 2: Identify values that are not in the quotient’s domain. Any values that result in division by zero, for example, cannot be substituted into the quotient. The domains of f(x) and g(x) are both made up entirely of real numbers, but we must never divide by zero while computing the quotient. If the following conditions exist:
2(x+4) = 0
(x+4) = 0
x = -4
As a result, the value x = -4 does not belong in the domain.
Step 3: To fully simplify the quotient, cancel any like factors. Step 2 values that were not in the domain are still not in the domain.
The common factor in the numerator and denominator is x + 4. By eliminating the common factor, we get at:
(f/g)(x) = [(x+7)(x+4)]/2(x+4)
(f/g)(x) = (x+7)/2
We still have x -4 due to the restrictions found in step 2.
Therefore,
(f/g)(x) = (x+7)/2 ; x ≠ -4
Conclusion
To find the derivative or differentiation of a function given as a ratio or division of two differentiable functions, one can use the calculus quotient rule. Because both f(x) and g(x) are differentiable, we can use the formula f(x) g(x) for the derivative of a function of the form f(x)/x, where g(x) > 0. After the product rule and differentiation’s concept of limits on derivation, the quotient rule follows directly on its heels.Previous to this, After reviewing the quotient formula and its applications in the previous chapter, we learnt how to use it to calculate the derivatives of a quotient of two differentiable functions, which was discussed in detail in the following chapter. Examine how to show the quotient rule formula in the following section.
