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Quotient Rule
In calculus, the quotient rule is a method of finding the derivative of a function (mathematics), function that is the ratio of two differentiable functions. Let h(x)=\frac, where both and are differentiable and g(x)\neq 0. The quotient rule states that the derivative of is :h'(x) = \frac. It is provable in many ways by using other Differentiation rules, derivative rules. Examples Example 1: Basic example Given h(x)=\frac, let f(x)=e^x, g(x)=x^2, then using the quotient rule:\begin \frac \left(\frac\right) &= \frac \\ &= \frac \\ &= \frac \\ &= \frac \\ &= \frac. \end Example 2: Derivative of tangent function The quotient rule can be used to find the derivative of \tan x = \frac as follows: \begin \frac \tan x &= \frac \left(\frac\right) \\ &= \frac \\ &= \frac \\ &= \frac \\ &= \frac = \sec^2 x. \end Reciprocal rule The reciprocal rule is a special case of the quotient rule in which the numerator f(x)=1. Applying the q ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Calculus
Calculus is the mathematics, mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizations of arithmetic operations. Originally called infinitesimal calculus or "the calculus of infinitesimals", it has two major branches, differential calculus and integral calculus. The former concerns instantaneous Rate of change (mathematics), rates of change, and the slopes of curves, while the latter concerns accumulation of quantities, and areas under or between curves. These two branches are related to each other by the fundamental theorem of calculus. They make use of the fundamental notions of convergence (mathematics), convergence of infinite sequences and Series (mathematics), infinite series to a well-defined limit (mathematics), limit. It is the "mathematical backbone" for dealing with problems where variables change with time or another reference variable. Infinitesimal calculus was formulated separately ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Derivative
In mathematics, the derivative is a fundamental tool that quantifies the sensitivity to change of a function's output with respect to its input. The derivative of a function of a single variable at a chosen input value, when it exists, is the slope of the tangent line to the graph of the function at that point. The tangent line is the best linear approximation of the function near that input value. For this reason, the derivative is often described as the instantaneous rate of change, the ratio of the instantaneous change in the dependent variable to that of the independent variable. The process of finding a derivative is called differentiation. There are multiple different notations for differentiation. '' Leibniz notation'', named after Gottfried Wilhelm Leibniz, is represented as the ratio of two differentials, whereas ''prime notation'' is written by adding a prime mark. Higher order notations represent repeated differentiation, and they are usually denoted in Leib ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Function (mathematics)
In mathematics, a function from a set (mathematics), set to a set assigns to each element of exactly one element of .; the words ''map'', ''mapping'', ''transformation'', ''correspondence'', and ''operator'' are sometimes used synonymously. The set is called the Domain of a function, domain of the function and the set is called the codomain of the function. Functions were originally the idealization of how a varying quantity depends on another quantity. For example, the position of a planet is a ''function'' of time. History of the function concept, Historically, the concept was elaborated with the infinitesimal calculus at the end of the 17th century, and, until the 19th century, the functions that were considered were differentiable function, differentiable (that is, they had a high degree of regularity). The concept of a function was formalized at the end of the 19th century in terms of set theory, and this greatly increased the possible applications of the concept. A f ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Differentiation Rules
This article is a summary of differentiation rules, that is, rules for computing the derivative of a function (mathematics), function in calculus. Elementary rules of differentiation Unless otherwise stated, all functions are functions of real numbers (\mathbb) that return real values, although, more generally, the formulas below apply wherever they are well defined, including the case of complex numbers (\mathbb). Constant term rule For any value of c, where c \in \mathbb, if f(x) is the constant function given by f(x) = c, then \frac = 0. Proof Let c \in \mathbb and f(x) = c. By the definition of the derivative: \begin f'(x) &= \lim_\frac \\ &= \lim_ \frac \\ &= \lim_ \frac \\ &= \lim_ 0 \\ &= 0. \end This computation shows that the derivative of any constant function is 0. Intuitive (geometric) explanation The derivative of the function at a point is the slope of the line tangent to the curve at the point. The slope of the constant function is 0, because the Tangent, tangen ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Chain Rule
In calculus, the chain rule is a formula that expresses the derivative of the Function composition, composition of two differentiable functions and in terms of the derivatives of and . More precisely, if h=f\circ g is the function such that h(x)=f(g(x)) for every , then the chain rule is, in Lagrange's notation, h'(x) = f'(g(x)) g'(x). or, equivalently, h'=(f\circ g)'=(f'\circ g)\cdot g'. The chain rule may also be expressed in Leibniz's notation. If a variable depends on the variable , which itself depends on the variable (that is, and are dependent variables), then depends on as well, via the intermediate variable . In this case, the chain rule is expressed as \frac = \frac \cdot \frac, and \left.\frac\_ = \left.\frac\_ \cdot \left. \frac\_ , for indicating at which points the derivatives have to be evaluated. In integral, integration, the counterpart to the chain rule is the substitution rule. Intuitive explanation Intuitively, the chain rule states that knowing t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Product Rule
In calculus, the product rule (or Leibniz rule or Leibniz product rule) is a formula used to find the derivatives of products of two or more functions. For two functions, it may be stated in Lagrange's notation as (u \cdot v)' = u ' \cdot v + u \cdot v' or in Leibniz's notation as \frac (u\cdot v) = \frac \cdot v + u \cdot \frac. The rule may be extended or generalized to products of three or more functions, to a rule for higher-order derivatives of a product, and to other contexts. Discovery Discovery of this rule is credited to Gottfried Leibniz, who demonstrated it using "infinitesimals" (a precursor to the modern differential). (However, J. M. Child, a translator of Leibniz's papers, argues that it is due to Isaac Barrow.) Here is Leibniz's argument: Let ''u'' and ''v'' be functions. Then ''d(uv)'' is the same thing as the difference between two successive ''uvs; let one of these be ''uv'', and the other ''u+du'' times ''v+dv''; then: \begin d(u\cdot v) & = (u + d ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Reciprocal Rule
In calculus, the reciprocal rule gives the derivative of the reciprocal of a function ''f'' in terms of the derivative of ''f''. The reciprocal rule can be used to show that the power rule holds for negative exponents if it has already been established for positive exponents. Also, one can readily deduce the quotient rule from the reciprocal rule and the product rule. The reciprocal rule states that if ''f'' is differentiable at a point ''x'' and ''f''(''x'') ≠ 0 then g(''x'') = 1/''f''(''x'') is also differentiable at ''x'' and g'(x) = -\frac. Proof This proof relies on the premise that f is differentiable at x, and on the theorem that f is then also necessarily continuous Continuity or continuous may refer to: Mathematics * Continuity (mathematics), the opposing concept to discreteness; common examples include ** Continuous probability distribution or random variable in probability and statistics ** Continuous ... there. Applying the definition ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Power Rule
In calculus, the power rule is used to differentiate functions of the form f(x) = x^r, whenever r is a real number. Since differentiation is a linear operation on the space of differentiable functions, polynomials can also be differentiated using this rule. The power rule underlies the Taylor series as it relates a power series with a function's derivatives. Statement of the power rule Let f be a function satisfying f(x)=x^r for all x, where r \in \mathbb. Then, :f'(x) = rx^ \, . The power rule for integration states that :\int\! x^r \, dx=\frac+C for any real number r \neq -1. It can be derived by inverting the power rule for differentiation. In this equation C is any constant. Proofs Proof for real exponents Let where r is any real number. If then where \ln is the natural logarithm function, or as was required. Therefore, applying the chain rule to we see that f'(x)=\frac e^= \fracx^r which simplifies to When we may use the same definition with where we now ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Absolute Value
In mathematics, the absolute value or modulus of a real number x, is the non-negative value without regard to its sign. Namely, , x, =x if x is a positive number, and , x, =-x if x is negative (in which case negating x makes -x positive), and For example, the absolute value of 3 and the absolute value of −3 is The absolute value of a number may be thought of as its distance from zero. Generalisations of the absolute value for real numbers occur in a wide variety of mathematical settings. For example, an absolute value is also defined for the complex numbers, the quaternions, ordered rings, fields and vector spaces. The absolute value is closely related to the notions of magnitude, distance, and norm in various mathematical and physical contexts. Terminology and notation In 1806, Jean-Robert Argand introduced the term ''module'', meaning ''unit of measure'' in French, specifically for the ''complex'' absolute value,Oxford English Dictionary, Draft Revision, Ju ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Natural Logarithm
The natural logarithm of a number is its logarithm to the base of a logarithm, base of the e (mathematical constant), mathematical constant , which is an Irrational number, irrational and Transcendental number, transcendental number approximately equal to . The natural logarithm of is generally written as , , or sometimes, if the base is implicit, simply . Parentheses are sometimes added for clarity, giving , , or . This is done particularly when the argument to the logarithm is not a single symbol, so as to prevent ambiguity. The natural logarithm of is the exponentiation, power to which would have to be raised to equal . For example, is , because . The natural logarithm of itself, , is , because , while the natural logarithm of is , since . The natural logarithm can be defined for any positive real number as the Integral, area under the curve from to (with the area being negative when ). The simplicity of this definition, which is matched in many other formulas ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Logarithmic Derivative
In mathematics, specifically in calculus and complex analysis, the logarithmic derivative of a function is defined by the formula \frac where is the derivative of . Intuitively, this is the infinitesimal relative change in ; that is, the infinitesimal absolute change in , namely scaled by the current value of . When is a function of a real variable , and takes real, strictly positive values, this is equal to the derivative of , or the natural logarithm of . This follows directly from the chain rule: \frac\ln f(x) = \frac \frac Basic properties Many properties of the real logarithm also apply to the logarithmic derivative, even when the function does ''not'' take values in the positive reals. For example, since the logarithm of a product is the sum of the logarithms of the factors, we have (\log uv)' = (\log u + \log v)' = (\log u)' + (\log v)' . So for positive-real-valued functions, the logarithmic derivative of a product is the sum of the logarithmic derivatives ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Logarithmic Differentiation
In calculus, logarithmic differentiation or differentiation by taking logarithms is a method used to differentiate functions by employing the logarithmic derivative of a function , (\ln f)' = \frac \quad \implies \quad f' = f \cdot (\ln f)'. The technique is often performed in cases where it is easier to differentiate the logarithm of a function rather than the function itself. This usually occurs in cases where the function of interest is composed of a product of a number of parts, so that a logarithmic transformation will turn it into a sum of separate parts (which is much easier to differentiate). It can also be useful when applied to functions raised to the power of variables or functions. Logarithmic differentiation relies on the chain rule as well as properties of logarithms (in particular, the natural logarithm, or the logarithm to the base '' e'') to transform products into sums and divisions into subtractions. The principle can be implemented, at least in part, in t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
