A formula for computing the indefinite integral of a quotient of two functions is known as the quotient rule of integration. The derivative of a function at a point may be found in calculus by taking the limit as delta x -> 0 on both sides of an equation using a difference quotient. What happens if we instead consider derivatives with respect to x? This is where the quotient rule comes into play.
As with any integration involving more than one variable, there are a variety of ways depending on the number of derivatives or anti-derivatives that must be taken and the conditions under which they must be taken. In general, it’s best to start with something for which there is an anti-derivative and then concentrate on simplifying the answer.
The following is how the quotient rule is written:
The functions “f”(“x”) and “g”(“x”) are differentiable functions, and “dx” is a very tiny integer with an integration constant C. Many integrals can be solved using this formula.
Anti-derivatives provide us with the exact same function as the original. However, now that this equation has been established, there are a few crucial restrictions to be aware of that can substantially simplify statements using derivatives or differentiation in practise. L’Hôpital’s Rule, which deals with roots of 0, Taylor’s Theorem on fractional powers derivation, and the mean value theorem, which allows one to build definite integrals, are all examples.
• L’Hôpital’s Rule: When a function has zero roots, the derivative must be limited in such a way that the roots have no effect on the outcome. To put it another way, if “g”(“x”) = “f”(“x”)/”h”(“x”). When this rule is applied,
1. f(x) is a polynomial, and g(x) is a rational expression with a denominator that does not approach 0 as x approaches infinity along any horizontal line in its domain (in other words, it only goes to zero at most once).
2. f(x) is an exponential or logarithmic function, and g(x) is an algebraic expression with the greatest level of differentiation in x equal to 1.
The result of the derivation must be multiplied by C in both circumstances.
• Taylor’s Theorem on Fractional Power Derivation: When a function possesses anti-derivatives that are fractional powers themselves, one can create definite integrals by using Taylor polynomials about 0. To put it another way, if “f”(“x”) = “g”(“x”)/”h”(“x”), the integral of “f”(“x”) is:
This result arises immediately from the fact that Taylor polynomials and higher powers of functions are equal, as stated by L’Hôpital’s Rule.
• Theorem of Mean Value: If, then one has. Because differentiation may not always be differentiable, this theorem derives directly from the Chain Rule and only applies to continuous functions over closed intervals. If f(x) = cos(ax)/a, for example, we get using the quotient rule. However, because there is no method to compute the derivative of a quotient using cosine at x=0, this is not differentiable.
However, using the geometric series formula with r=1/a, this integral may be calculated. These two formulas are equal when we substitute 0 for x in both situations, and they provide the same result.
In practice, knowing all of these limitations may substantially simplify some integrals, thus it’s crucial to know them all. If one has to find the area under the curve 1/(x2+1) on [0,1], for example, L’Hôpital’s Rule can be used twice. The outcome is a quarter.
The quotient rule is an important concept to understand when learning arithmetic in math classes. When the numerator and denominator of a function have previously been differentiated, the quotient rule is used to get the derivative.
