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October 4, 2021

The Magisterial Simplicity Of The Quotient Rule

The quotient rule of integration is a formula used to compute the indefinite integral of a quotient of two functions. In calculus, one can find the derivative of a function at a point by taking the limit as delta x -> 0 from both sides on an expression involving a difference quotient. One might ask what happens if we take derivatives with respect to x instead, and this is where the quotient rule comes into play.

 

As with any integration involving more than one variable, there are many possible approaches depending on how many derivatives or anti-derivatives need to be taken as well as under what conditions these may exist. In general, it is easiest to start with something for which an anti-derivative exists and then work on simplifying their solution.

 

The quotient rule is expressed as follows:

 

Here, the functions “f”(“x”) and “g”(“x”) are differentiable functions while “dx” is some very small number along with a constant of integration C. This formula can be used to solve many integrals.

 

Since taking anti-derivatives gives us back exactly the original function. However, now that this equation has been established, there are several important limits that arise in practice that are worth knowing about, which can greatly simplify expressions involving derivatives or differentiation. These include L’Hôpital’s Rule, which deals with roots of 0, Taylor’s Theorem on derivation by fractional powers, and the mean value theorem, which allows one’s to establish definite integrals.

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  • L’Hôpital’s Rule: When a function has roots of 0, one must limit the derivative in such a way that these roots will not affect the result. In other words, if “g”(“x”) = “f”(“x”)/”h”(“x”), then one must find . This rule applies when,

 

  1. f(x) is a polynomial, and g(x) is a rational expression whose denominator does not go to zero as x approaches infinity along any horizontal line in its domain (in other words, it only goes to zero at most once).

 

  1. f(x) is an exponential or logarithm function, and g(x) is an algebraic expression whose highest order of differentiation in x is 1.

In both cases, the result of the derivation must be multiplied by C.

 

  • Taylor’s Theorem on Derivation by Fractional Powers: When a function has anti-derivatives which are themselves fractional powers, one can establish definite integrals by taking advantage of Taylor polynomials around 0. In other words, if “f”(“x”) = “g”(“x”)/”h”(“x”), then the integral of “f”(“x”) is given by:

This result can be derived from L’Hôpital’s Rule and follows directly from the fact that Taylor polynomials and higher powers of functions are equivalent.

 

  • Mean Value Theorem: This theorem follows directly from the Chain Rule and only applies to continuous functions on closed intervals since differentiation may not always be differentiable. For example, if f(x) = cos(ax)/a, then applying the quotient rule gives us. However, this is not differentiable at x=0 because there is no way for us to find the derivative of a quotient involving cosine.
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However, this integral can be found by applying the geometric series formula with r=1/a to get. If we plug in 0 for x in both cases, these two expressions are equivalent and give us back what we originally wanted.

 

In practice, these limits can greatly simplify certain integrals, and it is important to be familiar with all of them. For example, if one needs to find the area under the curve 1/(x^2+1) on [0,1], he can use L’Hôpital’s Rule twice. The result is simply 1/4.

 

To learn math one should definitely know about quotient rule. The quotient rule is used to find the derivative of a function given that its numerator and denominator have already been differentiated.

 

Unlike the product rule, sum rule, and chain rule, all of which rely on implicit differentiation to find the derivative of a composite function, the quotient rule does not require implicit differentiation. Instead, it enables one to find derivatives through direct substitution. This makes it especially useful for finding derivatives involving trigonometric functions. These all become very easy to understand when one uses Cuemath. Cuemath provides much helpful and accessible information to the user.

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