AP Calculus AB review AP微积分复习提纲
AP Calculus AB review AP微积分复习提纲
AP CALCULUS AB REVIEWChapter 2DifferentiationDefinition of Tangent Line with Slop mIf f is defined on an open interval containing c, and if the limitexists, then the line passing through (c, f(c)) with slope mis the tangent line to the graph of f at the point (c, f(c)). Definition of the Derivative of a FunctionThe Derivative of f at x is given byprovided the limit exists. For all x for which this limit exists, f’ is a function of x.*The Power Rule*The Product Rule***The Chain Rule☺Implicit Differentiation (take the derivative on both sides;derivative of y is y*y’)Chapter 3Applications of Differentiation*Extrema and the first derivative test (minimum: − → + , maximum: + → − , + & − are the sign of f’(x) )*Definition of a Critical NumberLet f be defined at c. If f’(c) = 0 OR IF F IS NOT DIFFERENTIABLE AT C, then c is a critical number of f.*Rolle’s TheoremIf f is differentiable on the open interval (a, b) and f (a) = f (b), then there is at least one number c in (a, b) suchthat f’(c) = 0.*The Mean Value TheoremIf f is continuous on the closed interval [a, b] and differentiable on the open interval (a, b), then there existsa number c in (a, b) such that f’(c) = .*Increasing and decreasing interval of functions (take the first derivative)*Concavity (on the interval which f’’ > 0, concave up)*Second Derivative TestLet f be a function such that f’(c) = 0 and the second derivative of f exists on an open interval containing c.1.If f’’(c) > 0, then f(c) is a minimum2.If f’’(c) < 0, then f(c) is a maximum*Points of Inflection (take second derivative and set it equal to 0, solve the equation to get x and plug x value in originalfunction)*Asymptotes (horizontal and vertical)*Limits at Infinity*Curve Sketching (take first and second derivative, make sureall the characteristics of a function are clear)♫ Optimization Problems*Newton’s Method (used to approximate the zeros of a function,which is tedious and stupid, DO NOT HAVE TO KNOW IF U DO NOTWANT TO SCORE 5)Chapter 4 & 5Integration*Be able to solve a differential equation*Basic Integration Rules1)2)3)4)*Integral of a function is the area under the curve*Riemann Sum (divide interval into a lot of sub-intervals, calculate the area for each sub-interval and summation is theintegral).*Definite integral*The Fundamental Theorem of CalculusIf a function f is continuous on the closed interval [a, b]and F is an anti-derivative of f on the interval [a, b], then.*Definition of the Average Value of a Function on an Interval If f is integrable on the closed interval [a, b], then the average value of f on the interval is.*The second fundamental theorem of calculusIf f is continuous on an open internal I containing a, then,for every x in the interval,.*Integration by Substitution*Integration of Even and Odd Functions1) If f is an even function, then.2) If f is an odd function, then.*The Trapezoidal RuleLet f be continuous on [a, b]. The trapezoidal Rule forapproximating is given byMoreover, a n →∞, the right-hand side approaches.*Simpson’s Rule (n is even)Let f be continuous on [a, b]. Simpson’s Rule forapproximating isMoreover, as n→∞, the right-hand side approaches*Inverse functions(y=f(x), switch y and x, solve for x)*The Derivative of an Inverse FunctionLet f be a function that is differentiable on an intervalI. If f has an inverse function g, then g is differentiableat any x for which f’(g(x))≠0. Moreover,, f’(g(x))≠0.*The Derivative of the Natural Exponential FunctionLet u be a differentiable function of x.1. 2. .*Integration Rules for Exponential FunctionsLet u be a differentiable function of x..♠Derivatives for Bases other than eLet a be a positive real number (a ≠1) and let u be a differentiable function of x.1. 2.♠♠*Derivatives of Inverse Trigonometric FunctionsLet u be a differentiable function of x.*Definition of the Hyperbolic Functions。
2014年AP微积分CALCULUS知识点总结
A DERIV ATIVE FUNCTION1. The derivative function or simply the derivative is defined as)(x f '＝y '＝xx f x x f x y x x ∆-∆+=∆∆→∆→∆)()(lim lim002. Find the derivative function a) Find y ∆,b) Find the average rate of change x y ∆∆, c) Find the limit xy x ∆∆→∆0lim .3. Geometric significanceConsider a general function y=f(x), a fixed point A(a,f(a)) and a variable point B(x,f(x)). The slope of chord AB=ax a f x f --)()(.Now as B →A, x →a and the slope of chord AB →slope of tangent at A. So, ax a f x f a x --→)()(lim is )(a f '.Thus, we can know the derivative at x=a is the slope of the tangent at x=a.4. Rules)(x f)(x f 'C(a constant) 0n x1-n nxx sin x cosx cosx sin -x tanxx 22cos 1sec =x arcsin2-11x5. The chain ruleIf )(u f y = where )(x u u = thendxdu du dy dx dy =. )()(x g e x f = )()()(x g e x f x g '=' )(ln )(x g x f = )()()(x g x g x f '=' )(ln )()(ln )()()()(x u x v x u x v e e x u x f x v ===,])()()()(ln )([)()(ln )(x u x u x v x u x v ex f x u x v '+'='6. Inverse function, Parametric function and Implicit function Inverse function:dy dx dx dy 1=, ])([1)(1'='-x f x f , i.e., x y arcsin =, y x sin =Parametric function:dtdx dtdy dx dy =, i.e., )(t y ϕ=,)(t x ψ=→)(1x t -=ψ, )]([1x y -=ψϕ)()(t t dt dx dt dy dx dt dt dy dx dy ψϕ''=== Implicit function: 0))(,(=x y x F , 0))(,(=x f x F .0-222=+a y x ,ta y t a x sin cos ==, t ]2,0[π∈t ta t a dx dy x y cot sin cos )(-=-=='7. High derivativexx f x x f dx y d x f x ∆'-∆+'==''→∆)()(lim )(022 ta t a t dt dx dt y d dx y d x y x y x 32sin 1sin csc ])([)(-=-='='=''='' xx f x x f x f n n x n ∆-∆+=--→∆)()(lim )()1()1(0)( y=sinx )2sin(cos π+=='x x y , )22sin()2cos(ππ⨯+=+=''x x y )2sin()(π⨯+=n x ynB APPLICATIONS OF DIFFERENTIAL CALCULUS1. Monotonicitya) If S is an interval of real numbers and f(x) is defined for all x in S, then ：f(x) is increasing on S ⇔ 0)(≥'x f for all x in S, and f(x) is decreasing on S ⇔0)(≤'x f for all x in S. b) Find the monotone interval ● Find domain of the function,● Find )(x f ', and x which make 0)(='x f , ● Draw sign diagram, find the monotone interval. 2. Maxima/Minima, Horizontal inflection, Stationary pointC INTEGRAL1. The idea of definite integralWe define the unique number between all lower and upper sums as⎰badx x f )( and call it “the definite integral of )(x f from a to b ”,i.e., ∑∑⎰=-=∆〈〈∆ni i n i ba i x x f dx x f x x f 110)()()( where nab x -=∆.We note that as ∞→n ,∑⎰-=→∆10)()(n i ba idx x f x x f and⎰∑→∆=ba ni i dx x f x x f )()(1We write ⎰∑=∆=∞→ba ni i n dx x f x x f )()(lim 1. If 0)(≥x f for all x on [a,b] then⎰badx x f )( is the shaded area.2. Properties of definite integrals⎰⎰-=-bab adx x f dx x f )()]([⎰⎰=ba b a dx x f c dx x cf )()(, c is any constant ⎰⎰⎰=+ca ba cb dx x f dx x f dx x f )()()( ⎰⎰⎰+=+bababadx x g dx x f dx x g x f )()()]()([。
AP-Calculus-AB-review-AP微积分复习提纲
AP-Calculus-AB-review-AP微积分复习提纲AP CALCULUS AB REVIEWChapter 2DifferentiationDefinition of Tangent Line with Slop mIf f is defined on an open interval containing c, and if the limitlim ∆x→0∆y∆x=lim∆x→0f(c+∆x)−f(c)∆x=mexists, then the line passing through (c, f(c)) with slope m is the tangent line to the graph of f at the point (c, f(c)).Definition of the Derivative of a FunctionThe Derivative of f at x is given byf′(x)=lim∆x→0f(c+∆x)−f(c)∆xprovided the limit exists. For all x for which this limit exists, f’ is afunction of x.*The Power Rule*The Product Rule*ddx[sin x]=cos x*ddx[cos x]=−sin x*The Chain Rule☺Implicit Differentiation (take the derivative on both sides;derivative of y is y*y’)Chapter 3Applications of Differentiation*Extrema and the first derivative test (minimum: − → + , maximum:+ → −, + & − are the sign of f’(x) )*Definition of a Critical NumberLet f be defined at c. If f’(c) = 0 OR IF F IS NOTDIFFERENTIABLE AT C, then c is a critical number of f.1)∫u ndu =u n+1n+1+ C,n ≠−12)∫sin u du = −cos u + C 3)∫cos u du = sin u + C 4)∫1u du = ln u*Integral of a function is the area under the curve*Riemann Sum (divide interval into a lot of sub-intervals, calculatethe area for each sub-interval and summation is the integral).*Definite integral*The Fundamental Theorem of CalculusIf a function f is continuous on the closed interval [a, b] and F isan anti-derivative of f on the interval [a, b], then ∫f (x )dx ba=F (b )− F(a).*Definition of the Average Value of a Function on an IntervalIf f is integrable on the closed interval [a, b], then the averagevalue of f on the interval is 1b−a ∫f(x)dx ba.*The second fundamental theorem of calculusIf f is continuous on an open internal I containing a, then, for every x in the interval,ddx[∫f (t )dt xa ]=f(x).*Integration by Substitution*Integration of Even and Odd Functions1) If f is an even function, then ∫f (x )dx b a =2∫f(x)dx ba . 2) If f is an odd function, then ∫f (x )dx ba=0.*The Trapezoidal RuleLet f be continuous on [a, b]. The trapezoidal Rule forapproximating ∫f (x )dx bais given by ∫f (x )dx ba≈ b−a 2n[f (x 0)+2f (x 1)+2f (x 2)+⋯+2f (x n−1) +f (x n )]Moreover, an → ∞, the right-hand sideapproaches ∫f (x )dx ba. *Simpson ’s Rule (n is even)Let f be continuous on [a, b]. Simpson ’s Rule for approximating ∫f (x )dx bais ∫f (x )dx ba≈b −a3n [f (x 0)+4f (x 1)+2f (x 2)+4f (x 3)+⋯4f (x n−1)+f (x n )]Moreover, as n →∞, the right-hand side approaches ∫f (x )dx ba*Inverse functions(y=f (x), switch y and x, solve for x)*The Derivative of an Inverse FunctionLet f be a function that is differentiable on an interval I . If f has aninverse function g , then g is differentiable at any x for which f ’(g (x))≠0. Moreover, g ′(x )= 1f (g(x)), f ’(g (x))≠0.*The Derivative of the Natural Exponential Function Let u be a differentiable function of x .1.d dx[e x ]= e x 2.d dx[e u ]= e udu dx.*Integration Rules for Exponential Functions Let u be a differentiable function of x. ∫e u du = e u +C .♠Derivatives for Bases other than eLet a be a positive real number (a ≠1) and let u be adifferentiable function of x . 1.d dx[a u ]=(ln a)a udu dx2.ddx[log a u ]=1u ln a dudx♠∫a x dx =(1ln a)a x +C♠lim x→∞(1+1x)x =lim x→∞(x+1x)x=e*Derivatives of Inverse Trigonometric Functions Let u be a differentiable function of x. d dx [sin −1u ]=√2d dx[cos −1u ]=√2d dx[tan −1u ]=u′1+u=sin −1u a+C ∫du a 2+u 2=1atan −1ua+C=1a sec −1|u |a+C*Definition of the Hyperbolic Functions sinh x =e x −e −x2 cosh x =e x +e −x2tanh x =sinh x cosh x csch x =1sinh x ,x ≠0 sech x =1cosh x coth x =1tanh x,x ≠0。
(2020年整理)AP Calculus AB review AP微积分复习提纲.pptx
*The Derivative of the Natural Exponential Function Let u be a differentiable fu.
*Integration Rules for Exponential Functions Let u be a differentiable function of x.
*The Mean Value Theorem If f is continuous on the closed interval [a, b] and differentiable on the open interval (a, b), then there exists a number c in (a, b) such that f ’(c) =
area for each sub-interval and summation is the integral). *Definite integral *The Fundamental Theorem of Calculus
If a function f is continuous on the closed interval [a, b] and F is an anti-derivative of f on the interval [a, b], then
*Definition of the Hyperbolic Functions
学海无 涯
exists, then the line passing through (c, f(c)) with slope m is the tangent line to the graph of f at the point (c, f(c)). Definition of the Derivative of a Function The Derivative of f at x is given by
ap微积分大纲英文版
ap微积分大纲英文版The AP Calculus AB and BC course and exam description, published by the College Board, outlines the content and skills students are expected to master in an AP Calculus course. The document provides a detailed breakdown of the topics covered in both the AB and BC courses, as well asthe specific skills and knowledge students need to demonstrate in order to succeed on the AP exam.In the AP Calculus AB course, students are expected to develop a deep understanding of the concepts of limits, derivatives, integrals, and the fundamental theorem of calculus. They also study applications of derivatives and integrals, including but not limited to: analysis of graphs, optimization problems, and modeling with differential equations. Additionally, students are introduced to the concept of series and sequences.The AP Calculus BC course includes all the topics covered in the AB course, but in greater depth and with theaddition of several new topics. These include advanced techniques of integration, parametric, polar, and vector functions, as well as series and Taylor polynomials. The BC course also covers differential equations and the application of calculus to the physical sciences and engineering.In both courses, students are expected to develop a strong foundation in calculus and its applications, including the ability to work with functions represented in a variety of ways: graphical, numerical, analytical, and verbal. They are also expected to develop the skills necessary to justify and interpret results and solutions, and to use calculus to solve problems.The course and exam description provides a comprehensive outline of the content and skills that students need to master, as well as sample exam questions and scoring guidelines. It is an invaluable resource for both teachers and students as they prepare for the AP Calculus exam.。
AP微积分考试的知识点储备
AP微积分考试的知识点储备三立为大家整理了AP微积分考试的知识点储备的相关内容，供考生们参考，以下是详细内容。
1. AP微积分的预备知识AP微积分学习前，学生们应该掌握以下预备知识：(1)实数与数轴(初中知识)(2)绝对值(初中知识)(3)区间和邻域(高中知识)(4)函数的概念(自变量和因变量)、函数表示法(特别是图示法和解析法)、函数的定义域和值域、函数的几何特征：单调性、有界性、奇偶性、周期性。
(高中知识)(5)基本初等函数(常数函数、幂函数、指数函数、对数函数、三角函数和反三角函数)的表达式、定义域和图形。
(高中知识)(6)复合函数对于定义域和值域的理解(高中知识)(7)初等函数和隐函数的表示法和概念(高中知识)(8)数列的基本性质(高中知识)利用高中数学总复习资料可以帮助我们巩固微积分预备知识，国内大学财经类微积分课本的第一章一般会有对高中数学的简单回顾。
SAT1数学部分考的是代数、几何，相当于我国初中知识水平，SAT2数学部分主要包括函数、三角、几何。
SAT2数学分为数学一和数学二，其中数学一比较简单，数学二比较难，包括三角，矩阵，级数，向量和部分微积分。
由于SAT2数学二适用性更广泛，我国学生一般会选考SAT2数学二。
学生可以把准备SAT1数学部分和SAT2数学一和数学二考试的部分内容作为准备学习AP微积分和AP统计学的基础。
AP微积分基础主要在函数和三角。
AP统计学基础主要在概率。
2. AP微积分的学习和考试内容根据最新考试大纲规定的AP微积分的考试内容如下：第一部分：函数和极限(Functions and limits)(1)函数(Functions)(2)函数图像分析(Analysis of graphs)(3)函数的极限(包括单侧极限) (Limits of functions (including one-sided limits)(4)渐进和无穷(Asymptotic and unbounded behavior)(5)函数的连续性(Continuity as a property of functions)第二部分：导数(Derivatives)(1)导数的概念(Concept of the derivative)(2)在一个点处的导数(Derivative at a point)(3)导函数(包括中值定理等) (Derivative as a function)(4)二阶导数(Second derivatives)(5)导数的应用(Applications of derivatives)(6) 导数的运算(Computation of derivatives)第三部分：积分(Integrals)(1)定积分的概念和性质(Interpretations and properties of definite integrals)(2)积分的应用(Applications of integrals)(3)微积分基本定理(Fundamental Theorem of Calculus)(4)不定积分(Techniques of Antidifferentiation)(5)不定积分的应用( Applications of Antidifferentiation)(6)定积分的数值计算( Numerical approximations to definite integrals)第四部分：多项式估算和级数(Polynomial Approximations and Series)(1) 级数的定义(Concept of series)(2) 常数项级数(Series of constant terms)(3) 泰勒级数(Taylor series)注：微积分AB需要1年的课程学习时间，其内容大约占了美国大学一年的微积分课程内容的三分之二，而微积分BC需要1年多的课程学习时间，其内容包括了美国大学一年的微积分课程内容的全部。
ap课程预备微积分大纲
ap课程预备微积分大纲 AP课程预备微积分大纲
I. 概述
A. 课程目标
B. 课程要求
C. 考试信息
II. 函数与图像
A. 函数的定义与性质
B. 常见函数及其图像
C. 函数的变换与组合
III. 极限与连续性
A. 极限的概念与性质
B. 极限计算方法
C. 连续函数与间断点
IV. 导数与微分
A. 导数的定义与性质
B. 导数的计算方法
C. 微分的概念与应用
V. 微分应用
A. 最值与最优化
B. 函数的增减性与凹凸性
C. 泰勒级数与近似计算
VI. 积分与反导数
A. 定积分的定义与性质
B. 积分计算方法
C. 反导数与不定积分
VII. 积分应用
A. 曲线长度与曲面积
B. 微积分与物理学
C. 微积分与经济学
VIII. 微分方程
A. 常微分方程的概念与分类
B. 一阶微分方程的解法
C. 高阶微分方程的解法IX. 多元微积分
A. 多元函数的极限与连续性
B. 偏导数与方向导数
C. 多元函数的积分
X. 空间解析几何
A. 三维空间的坐标系与向量
B. 曲线与曲面方程
C. 空间曲线与曲面的参数化
以上是AP课程预备微积分的大纲，旨在为学生提供必要的基础知识和技能，以便更好地准备和应对AP微积分课程和考试。
大纲包括了函数与图像、极限与连续性、导数与微分、微分应用、积分与反导数、积分应用、微分方程、多元微积分和空间解析几何等内容。
通过系统性的学习和练习，学生将能够掌握微积分的基本概念、计算方法和应用技巧，为未来的学习打下坚实的基础。
ap微积分知识点
ap微积分知识点
AP微积分是高中阶段的一门课程，主要介绍微积分的基本概念和应用。
以下是一些AP微积分的知识点：
1. 导数：导数是函数在某一点的变化率，也可以理解为函数曲线在该点的切线斜率。
常见的导数计算法则包括求常数函数、幂函数、指数函数、对数函数、三角函数等的导数。
2. 微分：微分是导数的另一种表达方式，表示函数在某一点附近的近似线性变化量。
微分可以帮助我们研究函数的极值、曲线的凹凸性等性质。
3. 积分：积分是导数的逆运算，表示函数的累积效应。
通过积分可以计算曲线下的面积、变化量等。
常见的积分计算方法包括不定积分和定积分。
4. 不定积分：不定积分是求导的逆运算，表示函数的原函数。
不定积分的结果通常有一个常数项。
5. 定积分：定积分是计算函数在给定区间上的累积效应，表示曲线下的面积。
定积分可以通过反向求导的方式来计算。
6. 牛顿-莱布尼茨公式：牛顿-莱布尼茨公式是微积分的基本定理之一，它将积分和导数联系在一起。
该公式表明，函数的原函数与其在某一区间上的定积分之间存在关系。
7. 泰勒级数：泰勒级数是一种将函数展开成无穷级数的方法，可以用来近似表示复杂函数。
通过泰勒级数展开，我们可以研究函数的性质和计算函数的近似值。
以上是AP微积分的一些基本知识点，它们构成了微积分的核心内容。
掌握这些知识点能够帮助我们理解函数的变化规律、求解问题以及应用到实际生活中的各种情境中。
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AP CALCULUS AB REVIEWChapter 2DifferentiationDefinition of Tangent Line with Slop mIf f is defined on an open interval containing c, and if the limitlim ∆x→0∆y∆x=lim∆x→0f(c+∆x)−f(c)∆x=mexists, then the line passing through (c, f(c)) with slope m is the tangent line to the graph of f at the point (c, f(c)).Definition of the Derivative of a FunctionThe Derivative of f at x is given byf′(x)=lim∆x→0f(c+∆x)−f(c)∆xprovided the limit exists. For all x for which this limit exists, f’is afunction of x.*The Power Rule*The Product Rule*ddx[sin x]=cos x*ddx[cos x]=−sin x*The Chain Rule☺Implicit Differentiation (take the derivative on both sides; derivativeof y is y*y’)Chapter 3Applications of Differentiation*Extrema and the first derivative test (minimum: − → + , maximum: +→ −, + & − are the sign of f’(x) )*Definition of a Critical NumberLet f be defined at c. If f’(c) = 0 OR IF F IS NOT DIFFERENTIABLEAT C, then c is a critical number of f.*Rolle’s TheoremIf f is differentiable on the open interval (a, b) and f (a) = f (b), then thereis at least one number c in (a, b) such that f’(c) = 0.*The Mean Value TheoremIf f is continuous on the closed interval [a, b] and differentiable on theopen interval (a, b), then there exists a number c in (a, b) such that f’(c) =f(b)− f(a).b−a*Increasing and decreasing interval of functions (take the first derivative)*Concavity (on the interval which f’’ > 0, concave up)*Second Derivative TestLet f be a function such that f’(c) = 0 and the second derivative of f existson an open interval containing c.1.If f’’(c) > 0, then f(c) is a minimum2.If f’’(c) < 0, then f(c) is a maximum*Points of Inflection (take second derivative and set it equal to 0, solve theequation to get x and plug x value in original function)*Asymptotes (horizontal and vertical)*Limits at Infinity*Curve Sketching (take first and second derivative, make sure all thecharacteristics of a function are clear)♫ Optimization Problems*Newton’s Method (used to approximate the zeros of a function, which istedious and stupid, DO NOT HA VE TO KNOW IF U DO NOT WANTTO SCORE 5)Chapter 4 & 5Integration*Be able to solve a differential equation*Basic Integration Rules1)∫u ndu =u n+1n+1+ C,n ≠−12)∫sin u du = −cos u + C 3)∫cos u du = sin u + C 4)∫1u du = ln u*Integral of a function is the area under the curve*Riemann Sum (divide interval into a lot of sub-intervals, calculate thearea for each sub-interval and summation is the integral).*Definite integral*The Fundamental Theorem of CalculusIf a function f is continuous on the closed interval [a, b] and F is ananti-derivative of f on the interval [a, b], then ∫f (x )dx ba=F (b )− F(a).*Definition of the Average Value of a Function on an IntervalIf f is integrable on the closed interval [a, b], then the average value off on the interval is 1b−a ∫f(x)dx ba.*The second fundamental theorem of calculusIf f is continuous on an open internal I containing a, then, for every x in the interval,ddx[∫f (t )dt x a ]=f(x).*Integration by Substitution*Integration of Even and Odd Functions1) If f is an even function, then ∫f (x )dx b a =2∫f(x)dx ba . 2) If f is an odd function, then ∫f (x )dx ba=0.*The Trapezoidal RuleLet f be continuous on [a, b]. The trapezoidal Rule forapproximating ∫f (x )dx bais given by ∫f (x )dx ba≈ b−a 2n[f (x 0)+2f (x 1)+2f (x 2)+⋯+2f (x n−1) +f (x n )]Moreover, a n → ∞, the right-hand side approaches ∫f (x )dx ba. *Simpson ’s Rule (n is even)Let f be continuous on [a, b]. Simpson ’s Rule for approximating∫f (x )dx bais∫f (x )dx ba≈b −a3n [f (x 0)+4f (x 1)+2f (x 2)+4f (x 3)+⋯4f (x n−1)+f (x n )]Moreover, as n →∞, the right-hand side approaches∫f (x )dx ba*Inverse functions(y=f (x), switch y and x, solve for x)*The Derivative of an Inverse FunctionLet f be a function that is differentiable on an interval I . If f has aninverse function g , then g is differentiable at any x for which f ’(g (x))≠0. Moreover, g ′(x )=1f ′(g(x)), f ’(g (x))≠0.*The Derivative of the Natural Exponential Function Let u be a differentiable function of x .1.d dx[e x ]= e x 2.d dx[e u ]= e udu dx.*Integration Rules for Exponential Functions Let u be a differentiable function of x. ∫e u du = e u +C .♠Derivatives for Bases other than eLet a be a positive real number (a ≠1) and let u be a differentiablefunction of x . 1.d dx[a u ]=(ln a)a udu dx2.ddx[log a u ]=1u ln a dudx♠∫a x dx =(1ln a)a x +C ♠lim x→∞(1+1x)x =lim x→∞(x+1x)x=e*Derivatives of Inverse Trigonometric Functions Let u be a differentiable function of x.d dx [sin−1u]=√2ddx[cos−1u]=√2d dx [tan−1u]=u′1+u2√22=sin−1ua+C∫dua2+u2=1atan−1ua+C√22=1asec−1|u|a+C*Definition of the Hyperbolic Functionssinh x=e x−e−x2cosh x=ex+e−x2tanh x=sinh xcosh x csch x=1sinh x,x≠0sech x=1cosh x coth x=1tanh x,x≠0。