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# 数学代写|微积分代写Calculus代考|MATH1051 The Derivative

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## 数学代写|微积分代写Calculus代考|The Derivative

Suppose that $f$ is a function whose domain contains the interval $(a, b)$. Let $c$ be a point of $(a, b)$. If the limit
$$\lim {h \rightarrow 0} \frac{f(c+h)-f(c)}{h}$$ exists then we say that $f$ is differentiable at $c$ and we call the limit the derivative of $f$ at $c$. EXAMPLE 2.10 Is the function $f(x)=x^2+x$ differentiable at $x=2$ ? If it is, calculate the derivative. SOLUTION We calculate the limit $(*)$, with the role of $c$ played by 2 : \begin{aligned} \lim {h \rightarrow 0} \frac{f(2+h)-f(2)}{h} &=\lim {h \rightarrow 0} \frac{\left[(2+h)^2+(2+h)\right]-\left[2^2+2\right]}{h} \ &=\lim {h \rightarrow 0} \frac{\left[\left(4+4 h+h^2\right)+(2+h)\right]-[6]}{h} \ &=\lim {h \rightarrow 0} \frac{5 h+h^2}{h} \ &=\lim {h \rightarrow 0} 5+h \ &=5 . \end{aligned}

## 数学代写|微积分代写Calculus代考|Rules for Calculating Derivatives

Calculus is a powerful tool, for much of the physical world that we wish to analyze is best understood in terms of rates of change. It becomes even more powerful when we can find some simple rules that enable us to calculate derivatives quickly and easily. This section is devoted to that topic.

I Derivative of a Sum [The Sum Rule]: We calculate the derivative of a sum (or difference) by
$$(f(x) \pm g(x))^{\prime}=f^{\prime}(x) \pm g^{\prime}(x) .$$
In our many examples, we have used this fact implicitly. We are now just enunciating it formally.

II Derivative of a Product [The Product Rule]: We calculate the derivative of a product by
$$[f(x) \cdot g(x)]^{\prime}=f^{\prime}(x) \cdot g(x)+f(x) \cdot g^{\prime}(x) .$$
We urge the reader to test this formula on functions that we have worked with before. It has a surprising form. Note in particular that it is not the case that $[f(x) \cdot g(x)]^{\prime}=f^{\prime}(x) \cdot g^{\prime}(x)$.

III Derivative of a Quotient [The Quotient Rule]: We calculate the derivative of a quotient by
$$\left[\frac{f(x)}{g(x)}\right]^{\prime}=\frac{g(x) \cdot f^{\prime}(x)-f(x) \cdot g^{\prime}(x)}{g^2(x)} .$$
In fact one can derive this new formula by applying the product formula to $g(x) \cdot[f(x) / g(x)]$. We leave the details for the interested reader.

IV Derivative of a Composition [The Chain Rule]: We calculate the derivative of a composition by
$$[f \circ g(x)]^{\prime}=f^{\prime}(g(x)) \cdot g^{\prime}(x) .$$
To make optimum use of these four new formulas, we need a library of functions to which to apply them.

## 数学代写|微积分代写Calculus代考|The Derivative

$$\lim h \rightarrow 0 \frac{f(c+h)-f(c)}{h}$$

$$\lim h \rightarrow 0 \frac{f(2+h)-f(2)}{h}=\lim h \rightarrow 0 \frac{\left[(2+h)^2+(2+h)\right]-\left[2^2+2\right]}{h} \quad=\lim h \rightarrow 0 \frac{\left[\left(4+4 h+h^2\right)+(2+h)\right]-[6]}{h}=\lim h \rightarrow 0 \frac{5 h+h^2}{h}$$

## 数学代写|微积分代与Calculus代考|Rules for Calculating Derivatives

1和的导数 [求和规则]：我们计算和 (或差) 的导数
$$(f(x) \pm g(x))^{\prime}=f^{\prime}(x) \pm g^{\prime}(x) .$$

II 产品的导数[产品规则]：我们通过以下方式计算产品的导数
$$[f(x) \cdot g(x)]^{\prime}=f^{\prime}(x) \cdot g(x)+f(x) \cdot g^{\prime}(x) .$$

III 商的导数[商规则]：我们通过以下公式计算商的导数
$$\left[\frac{f(x)}{g(x)}\right]^{\prime}=\frac{g(x) \cdot f^{\prime}(x)-f(x) \cdot g^{\prime}(x)}{g^2(x)} .$$

IV 组合物的导数[链式法则]: 我们通过以下方式计算组合物的导数
$$[f \circ g(x)]^{\prime}=f^{\prime}(g(x)) \cdot g^{\prime}(x) .$$

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。