The Derivate

The derivate of a function $f(x)$ with respect to its variable $x$ is the instantaneous rate of change (slope) of the function at any value of $x$ .

$\begin{tikzpicture}[domain=0:4] % grid \draw[help lines] (0,0) grid (4,4); % origin \draw[red, line width=.1mm] (-0.1,-0.1) -- (0.1,0.1) (0.1,-0.1) -- (-0.1,0.1); \coordinate[label={[red]below:$O$}] (O) at (0,0); \draw[dotted,->] (O) -- (4,0) node[below] {$x$}; \draw[dotted,->] (O) -- (0,4) node[left] {$y$}; \draw[color=blue] plot[id=exp] function{0.05*exp(x)} node[right] {$y = f(x) = \frac{1}{20} \mathrm e^x$}; \coordinate[label={[red]below:$f(x_{1})$}] (F1) at (2, {1/20*exp(2)}); \drawpoint{F1}{.5mm}[red]; \coordinate[label={[red]above:$f(x_{2})$}] (F2) at (3.5, {1/20*exp(3.5)}); \drawpoint{F2}{.5mm}[red]; \draw[green] (F1) -- (3.5,{1/20*exp(2)}) node [below left] {$\Delta x$} -- (F2) node [below right] {$\Delta f$}; \end{tikzpicture}$

$\begin{eqnarray*} \frac{dy}{dx} &=& \lim_{\Delta x \to 0} \frac{\Delta f}{\Delta x} \end{eqnarray*}$

Derivates of Common Functions

${\color{gray}a}$ is a constant, while ${\color{red}u}$ , ${\color{blue}v}$ , ${\color{green}w}$ are functions. The derivates are expressed as derivatives with respect to an arbitrary variable x.

$\begin{eqnarray*} \frac{d}{d{\bf{x}}}{\color{gray}a} &=& 0 & \\ \frac{d}{d{\bf{x}}}{\bf{x}} &=& 1 & \\ \frac{d}{d{\bf{x}}}{\color{gray}a}{\color{red}u} &=& {\color{gray}a}\frac{d{\color{red}u}}{d{\bf{x}}} & \\ \frac{d}{d{\bf{x}}}{\color{red}u}+{\color{blue}v}-{\color{green}w} &=& \frac{d{\color{red}u}}{d{\bf{x}}} + \frac{d{\color{blue}v}}{d{\bf{x}}} - \frac{d{\color{green}w}}{d{\bf{x}}} & \\ \frac{d}{d{\bf{x}}}{\color{red}u}{\color{blue}v} &=& {\color{red}u}\frac{d{\color{blue}v}}{d{\bf{x}}}+{\color{blue}v}\frac{d{\color{red}u}}{d{\bf{x}}} & \\ \frac{d}{d{\bf{x}}}\frac{{\color{red}u}}{{\color{blue}v}} &=& \frac{1}{{\color{blue}v}}\frac{d{\color{red}u}}{d{\bf{x}}}-\frac{{\color{red}u}}{{\color{blue}v}^{2}}\frac{d{\color{blue}v}}{d{\bf{x}}} & \\ \frac{d}{d{\bf{x}}}{\color{red}u}^n &=& nu^{n-1}\frac{d{\color{red}u}}{d{\bf{x}}} & \\ \frac{d}{d{\bf{x}}}\sqrt{{\color{red}u}} &=& \frac{1}{2\sqrt{{\color{red}u}}}\frac{d{\color{red}u}}{d{\bf{x}}} & \\ \end{eqnarray*}$

$\begin{eqnarray*} \frac{d}{d{\bf{x}}}\frac{1}{{\color{red}u}} &=& -\frac{1}{{\color{red}u}^2}\frac{d{\color{red}u}}{d{\bf{x}}} & \\ \frac{d}{d{\bf{x}}}\frac{1}{{\color{red}u}^{n}} &=& -\frac{n}{{\color{red}u}^{n+1}}\frac{d{\color{red}u}}{d{\bf{x}}} & \\ \frac{d}{d{\bf{x}}}[f({\color{red}u})] &=& \frac{d}{d{\color{red}u}}[f({\color{red}u})]\frac{d{\color{red}u}}{d{\bf{x}}} & \\ \frac{d}{d{\bf{x}}}[\ln{{\color{red}u}}] &=& \frac{d}{d{\bf{x}}}[\log_{e}{{\color{red}u}}]=\frac{1}{{\color{red}u}}\frac{d{\color{red}u}}{d{\bf{x}}} & \\ \frac{d}{d{\bf{x}}}[\log_{{\color{gray}a}}{\color{red}u}] &=& \log_{{\color{gray}a}}{e}\frac{1}{{\color{red}u}}\frac{d{\color{red}u}}{d{\bf{x}}} & \\ \frac{d}{d{\bf{x}}}e^{{\color{red}u}} &=& e^{{\color{red}u}}\frac{d{\color{red}u}}{d{\bf{x}}} & \\ \frac{d}{d{\bf{x}}}{\color{gray}a}^{{\color{red}u}} &=& {\color{gray}a}^{{\color{red}u}}\ln{{\color{gray}a}}\frac{d{\color{red}u}}{d{\bf{x}}} & \\ \frac{d}{d{\bf{x}}}{\color{red}u}^{{\color{blue}v}} &=& {\color{blue}v}{\color{red}u}^{{\color{blue}v}-1}\frac{d{\color{red}u}}{d{\bf{x}}}+{\color{red}u}^{{\color{blue}v}}\ln{{\color{red}u}}\frac{d{\color{blue}v}}{d{\bf{x}}} & \\ \end{eqnarray*}$

$\begin{eqnarray*} \frac{d}{d{\bf{x}}}\sin{{\color{red}u}} &=& \cos{{\color{red}u}}\frac{d{\color{red}u}}{d{\bf{x}}} & \\ \frac{d}{d{\bf{x}}}\cos{{\color{red}u}} &=& -\sin{{\color{red}u}}\frac{d{\color{red}u}}{d{\bf{x}}} & \\ \frac{d}{d{\bf{x}}}\tan{{\color{red}u}} &=& \sec^{2}{{\color{red}u}}\frac{d{\color{red}u}}{d{\bf{x}}} & \\ \frac{d}{d{\bf{x}}}\cot{{\color{red}u}} &=& -\csc^2{{\color{red}u}}\frac{d{\color{red}u}}{d{\bf{x}}} & \\ \frac{d}{d{\bf{x}}}\sec{{\color{red}u}} &=& \sec{{\color{red}u}}\tan{{\color{red}u}}\frac{d{\color{red}u}}{d{\bf{x}}} & \\ \frac{d}{d{\bf{x}}}\csc{{\color{red}u}} &=& -\csc{{\color{red}u}}\cot{{\color{red}u}}\frac{d{\color{red}u}}{d{\bf{x}}} & \\ \end{eqnarray*}$

Partial Derivates

When there is more than one variable in a function it is often useful to examine the variation of the function with respect to one of the variables with all the other variables considered to stay constant.

Suppose we have a function:

$\begin{eqnarray*} F(x,y) &=& ax^{2} + by^{2} \end{eqnarray*}$

then, the partial derivate with respect to $x$ is written as:

$\begin{eqnarray*} \frac{\partial{F(x,y)}}{\partial{x}} &=& \frac{\partial}{\partial{x}}[ax^2+by^2] = 2ax \end{eqnarray*}$

respectively, after deriving by $y$ :

$\begin{eqnarray*} \frac{\partial{F(x,y)}}{\partial{y}} &=& \frac{\partial}{\partial{y}}[ax^2+by^2] = 2by \end{eqnarray*}$

Practical Applications

In physics, partial derivates are used to express various equations that rely on time. For example, the velocity $v$ is the derivate of the position with respect to time:

$\begin{eqnarray*} v &=& \frac{dp}{dt} \end{eqnarray*}$

and acceleration is the derivative of velocity with respect to time (and hence the double derivate):

$\begin{eqnarray*} v &=& \frac{dp}{dt} \Rightarrow & \\ a &=& \frac{d^{2}p}{dt^{2}} \end{eqnarray*}$

Combination of Functions

When applied to combination of functions certain properties of derivates apply.

Product of Two Functions

If a function $f(x)$ is a product of two functions $m(x)$ and $n(x)$ , then the derivate of this product is:

$\begin{equation*} \frac{df(x)}{dx} &=& \frac{d(m(x)n(x))}{dx} &=& m\frac{dn}{dx} + n\frac{dm}{dx} \end{equation*}$

Sum of Two Functions

If the function $f(x)$ is a sum of two functions $m(x)$ and $n(x)$ , then the derivate of this sum is:

$\begin{equation*} \frac{df(x)}{dx} &=& \frac{d(m(x) + n(x))}{dx} &=& \frac{dn}{dx} + \frac{dm}{dx} \end{equation*}$

Chain Rule

If the function $y=f(x)$ and $x=g(z)$ , then the derivative of $y$ with respect to $z$ can be written as a product of derivates:

$\begin{equation*} \frac{df(x)}{dz} &=& \frac{d(f(x))}{dx}\frac{dx}{dz} \end{equation*}$

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