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Topic: Sum rule in differentiation

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	Karl's Calculus Tutor - Box 4.4x: Rules for Derivatives
		Rule 3) The derivative of the sum is the sum of the derivatives.
		Rule 6) To find the derivative of a quotient or ratio, take the denominator times the derivative of the numerator, subtract from it the numerator times the derivative of the denominator, then divide the whole thing by the square of the denominator.
		Rule 9) If you add a constant to the independent variable, just treat the sum of the two as if it were the independent variable itself.
	www.karlscalculus.org /divrules.html (851 words)

	Sum rule in differentiation: Definition and Links by Encyclopedian.com
		The sum rule in differentiation is possibly the most useful rule in differentiation.
		The rule itself is a direct consequence of differentiation from first principles[?].
		The sum rule in differentiation can be used as part of the derivation for both the sum rule in integration and linearity of differentiation.
	www.encyclopedian.com /su/Sum-rule-in-differentiation.html (370 words)

	Linearity of differentiation - Wikipedia, the free encyclopedia
		In mathematics, the linearity of differentiation is a most fundamental property of the derivative, in differential calculus.
		It follows from the sum rule in differentiation and the constant factor rule in differentiation.
		By the constant factor rule in differentiation, this reduces to:
	en.wikipedia.org /wiki/Linearity_of_differentiation (128 words)

	Calculus
		The chain rule for differentiation of composite functions is covered and examples are covered illustrating how and when to apply the chain rule.
		This segment covers the exponential and logarithmic functions as well as their derivatives, and techniques for differentiation of functions involving the exponential and logarithmic functions using differentiation rules.
		The coordinate formulas for the distance between a pair of points in space, the normalization of a vector, the equation of a sphere, and the distance from a point to a plane are demonstrated using vectors.
	www.tutorace.com /html/calculus1.html (2054 words)

	Derivative Summary
		Rules 4 and 6 are special cases of Rules 5 and 7, respectively, when a = e.
		However, Rules 4 and 6 are simpler and the "change of base" technique used in the derivation of Rules 5 and 7 is useful to know.
		Derivative of the sum of functions is the sum of the derivatives, i.e.
	www.ac.aup.fr /~kdunz/Courses/Calculus/DSummary.htm (828 words)

	Math 162 - Class Summaries Page 1
		We worked out the differentiation formula for tanh(x) in class using the Quotient Rule, the differentiation formulas for sinh(x) and cosh(x), and the Fundamental Hyperbolic Identity; for homework you will repeat this kind of argumnt for sech(x), coth(x) and csch(x).
		The class ended with a proof of this rule that was simplified by making the extra assumption that the functions in the numerator and denominator of the original limit had derivatives that were continuous at the point where the limit was being computed.
		An example, the differentiation of 2^x, was worked at this point as a reminder that the idea of converting powers into exponentials was something that had already been introduced in first semester calculus.
	public.csusm.edu /public_html/DJBarskyWebs/162summarypage1.html (3788 words)

	Rules of Differentiation: The Sum/Difference Rule (Site not responding. Last check: 2007-09-10)
		These rules are simply formulas that instruct the learner how to compute derivatives depending on a given function.
		A useful rule of differentiation is the sum/difference rule.
		This rule simply tells us that the derivative of the sum/difference of functions is the sum/difference of the derivatives.
	jwilson.coe.uga.edu /EMAT6680/Horst/derivativesum/derivativesum.html (70 words)

	Implicit Differentiation
		Use the sum rule of differentiation to the whole term on the left of the given equation.
		Note that in calculating d [siny] / dx, we used the chain rule since y is itself a function of x and sin (y) is a function of a function.
		Use the differentiation of a sum formula to left side of the given equation.
	www.analyzemath.com /calculus/Differentiation/implicit.html (372 words)

	Rules of calculus - functions of one variable
		The rules of differentiation are cumulative, in the sense that the more parts a function has, the more rules that have to be applied.
		The power rule combined with the coefficient rule is used as follows: pull out the coefficient, multiply it by the power of x, then multiply that term by x, carried to the power of n - 1.
		Read this rule as: if y is equal to the sum of two terms or functions, both of which depend upon x, then the function of the slope is equal to the sum of the derivatives of the two terms.
	www.columbia.edu /itc/sipa/math/calc_rules_func_var.html (2396 words)

	Some notes for Quantitative Methods, lecture 8
		The short answer is that when you differentiate a function f(x), you get a new function of x, written df/dx, which gives the slope or gradient of the function at point x.
		There are altogether four rules we need for differentiation and this is the first of them.
		We now use the sum rule, and the definition of the tangent, to calculate the equation of the tangent to the curve y = f(x) = 2 x^2 + 3 x at the point x = 1.
	www.ee.surrey.ac.uk /Personal/J.Deane/Teach/qm1/lecs/lec8.html (800 words)

	SparkNotes: Computing Derivatives: Techniques of Differentiation
		In this section, we introduce the basic techniques of differentiation and apply them to functions built up from the elementary functions.
		In words, these properties say that the derivative of a constant times a function is that constant times the derivative of the function, and the derivative of a sum of functions is the sum of the derivatives of the functions.
		The idea is to differentiate both sides of the equation with respect to x (using the chain rule where necessary).
	www.sparknotes.com /math/calcbc1/computingderivatives/section2.rhtml (769 words)

	Example: Symbolic Differentiation
		That is, to obtain the derivative of a sum we first find the derivatives of the terms and add them.
		For a sum, for example we want to be able to extract the addend (first term) and the augend (second term).
		Since the differentiation program is defined in terms of abstract data, we can modify it to work with different representations of expressions solely by changing the predicates, selectors, and constructors that define the representation of the algebraic expressions on which the differentiator is to operate.
	mitpress.mit.edu /sicp/full-text/sicp/book/node39.html (1185 words)

	Math1b, Summer 2003, Introduction to Functions and Calculus II
		Differential equations permeate quantitative analysis throughout the sciences (in physics, chemistry, biology, enviromental science, astronomy) and social sciences.
		By the end of the course you will appreciate the power and usefulness differential equations and you will see how the work we have done with both series and integration comes into play in analyzing their solutions.
		For this reason the process of slicing, approximating a quantity on a slice, summing over all the slices to get a Riemann Sum, and taking the appropriate limit is the real heart of the applications section.
	www.math.harvard.edu /archive/1b_summer_03/syllabus.html (1685 words)

	Kenny's Consummate Clues for Calculus Class (Site not responding. Last check: 2007-09-10)
		WARNING: when using L'Hopital's Rule, the separate derivatives must be taken, it is not the derivative of the rational function, but the two separate derivatives of the numerator and denominator.
		This method is intimately related to the chain rule for differentiation.
		This is an illustration of the chain rule "backwards".
	coweb.math.gatech.edu:8888 /calculus/1616 (927 words)

	Differentiation
		All the results that we explain in this section can be proven using the first principles, which has been explained in the previous section.
		This rule forms the base for differentiating all functions.
		Now, armed with the basic knowledge of differentiation, go forth and explore more.
	library.thinkquest.org /C0110248/calculus/difnintro.htm (95 words)

	The Derivative
		The derivative is one of the pivotal concepts in 'differential calculus.' The derivative, in it's most basic form, is a slope predictor for a function.
		The first rule that is particularly useful for differentiating polynomial equations is the 'General Power Rule'.
		The sum rule is almost superfluous, though it is defined.
	www.calistoga.k12.ca.us /chs/Student_WebPages/Class_of_2000/Adam_Coates/MathAnalysis/chapters/derivative.htm (1006 words)

	3. The Sum Rule
		The proof, justification and further explanation of rules for differentiation may be found in this site area.
		But we will also introduce rules of differentiation which permit the calculation of formulas for f '(x) from formulas for f(x), calculation shortcuts for the evaluation of the limit definition of f'(x).
		Then rules are developed to evaluate the limit directly or replace the limit evaluation by an equivalent calculation in which there is no mention of limits.
	www.whyslopes.com /Calculus-Introduction/Differentiation-Sum-Rule.html (506 words)

	Summary: Techniques of Differentiation
		If the CTE says, for instance, that the expression is a sum of two smaller expressions, then apply the rule for sums as a first step.
		The following table summarizes the derivatives of logarithmic and exponential functions, as well as their chain rule counterparts (that is, the logarithmic and exponential functions of a function).
		Logarithmic differentiation is a useful alternative to the product and quotient rules when finding derivatives of particularly complicated expressions.
	www.zweigmedia.com /ThirdEdSite/Calcsumm4.html (697 words)

	2.8 Differentiation Rules
		If there had not been easily applied rules for finding the derivative of most functions used in modeling, the derivative would not be as powerful a tool as it has turned out to be.
		Using the limit definition, general rules are developed for constant multiples of a function; sums, products, reciprocals, quotients, and compositions of functions.
		Differentiability implies continuity, and an example showing that the converse is false.
	www.math.dartmouth.edu /~klbooksite/2.08/208.html (349 words)

	Math 134, Summary of Test #2 Lectures (Site not responding. Last check: 2007-09-10)
		the derivative of a sum is the sum of the derivatives
		Implicit Differentiation: Occasionally problems arise in which the function to be analyzed is "hidden" in an equation.
		This is the reason for the Rule of 70, i.e., to find the time needed for an investment to double in value, divide 70 by the percent interest being earned.
	www.math.uiuc.edu /~dcmurphy/math134/summary2.html (1364 words)

	False mathematical proofs@Everything2.com
		Since we are trying to perform a continuous operation (differentiation) on something defined discretely, we are committing a major error.
		A neater example of this trick, given explicitly in proof form, involves "proving" that a particular triangle has more than 180 degrees in its three angles, a violation of the triangle sum rule.
		But this means that triangle AHK has two 90 degree angles, an impossibility, since the third angle is yet unaccounted for and every triangle must have exactly 180 degrees by the Triangle Sum Rule.
	www.everything2.com /?node_id=1195306 (2138 words)

	[No title]
		This says that the derivative of a (ﬁnite) sum is the sum of the derivatives of the summands.
		These can be very simply differentiated by using the product rule alone, resulting in differentiation rules for radicals and quotients.
		Integration is a harder problem than differentiation, since the only procedure for performing integrals is to either do the Riemann sum directly, or ﬁnd a function whose derivative is the integrand, or to perform variable changes that put the integrand into a more readily recognized form.
	rustam.uwp.edu /CALC/calculus.html (1891 words)

	Sum rule in differentiation - Wikipedia, the free encyclopedia
		In calculus, the sum rule in differentiation is a method of finding the derivative of a function that is the sum of two other functions for which derivatives exist.
		This rule also applies to subtraction and to additions and subtractions of more than two functions
		Now use the special case of the constant factor rule in differentiation with k=−1 to obtain:
	en.wikipedia.org /wiki/Sum_rule_in_differentiation (292 words)

	Courses
		A survey of differential and integral calculus for students in the behavioral, life and social sciences.
		Second and higher order differential equations; homogeneous equations with constant coefficients, characteristics equations and their roots, homogeneous Euler type equations.
		A first course dealing with basic numerical methods for finding roots of non linear equations, interpolation theory, approximation of functions, numerical integration and differentiation, numerical solutions of systems of linear equations, the matrix eigen value problem and the numerical solutions of ordinary differential equations.
	www.bridgeport.edu /pages/3624.asp (1153 words)

	[No title]
		In this case, the function is not differentiable at the given point.
		In this case, the function is not differentiable at the given point but the limit of the symmetric difference quotient exists.
		Using the TI85 graphing calculator to plot approximating the graph of a derivative with the Newton quotient.
	archives.math.utk.edu /visual.calculus/2/index.html (754 words)

	The Citizen Scientist
		This tells us that the derivative of a sum is the sum of the derivatives.
		There is a presentation of many of the principles of differentiation (like the sum rule and so forth.) These are applied to many different types of functions.
		The chain rule is presented and then the idea of implicit differentiation is explored.
	www.sas.org /tcs/weeklyIssues_2005/2005-03-11/mathcorner/index.html (645 words)

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