Calculus Course/Differentiation

Derivative

A derivative is a mathematical operation to find the rate of change of a function.

Formula

For a non linear function $f(x)=y$ . The rate of change of $f(x)$ correspond to change of $x$ is equal to the ratio of change in $f(x)$ over change in $x$

$\frac{\mathrm{\Delta }f(x)}{\mathrm{\Delta }x}=\frac{\mathrm{\Delta }y}{\mathrm{\Delta }x}$

Then the Derivative of the function is defined as

$\frac{d}{dx}f(x)=\underset{\mathrm{\Delta }x\to 0}{\lim }\sum \frac{\mathrm{\Delta }f(x)}{\mathrm{\Delta }x}=\underset{\mathrm{\Delta }x\to 0}{\lim }\sum \frac{y}{x}$

but the derivative must exist uniquely at the point x. Seemingly well-behaved functions might not have derivatives at certain points. As examples, $f(x)=\frac{1}{x}$ has no derivative at $x=0$ ; $F(x)=|x|$ has two possible results at $x=0$ (-1 for any value for which $x<0$ and 1 for any value for which $x>0$) On the other side, a function might have no value at $x$ but a derivative of $x$ , for example $f(x)=\frac{x}{x}$ at $x=0$ . The function is undefined at $x=0$ , but the derivative is 0 at $x=0$ as for any other value of $x$ .

Practically all rules result, directly or indirectly, from a generalized treatment of the function.

Table of Derivative

General Rules

$\frac{d}{dx}(f+g)=\frac{df}{dx}+\frac{dg}{dx}$

$\frac{d}{dx}(c\cdot f)=c\cdot \frac{df}{dx}$

$\frac{d}{dx}(f\cdot g)=f\cdot \frac{dg}{dx}+g\cdot \frac{df}{dx}$

$\frac{d}{dx}\left(\frac{f}{g}\right)=\frac{g\cdot \frac{df}{dx}-f\cdot \frac{dg}{dx}}{g^2}$

Powers and Polynomials

$\frac{d}{dx}(c)=0$

$\frac{d}{dx}x=1$

$\frac{d}{dx}{x}^n=n{x}^{n-1}$

$\frac{d}{dx}\sqrt{x}=\frac{1}{2\sqrt{x}}$

$\frac{d}{dx}\frac{1}{x}=-\frac{1}{x^2}$

$\frac{d}{dx}(c_n{x}^n+c_{n-1}{x}^{n-1}+c_{n-2}{x}^{n-2}+\cdots +c_2{x}^2+c_{1}x+c_0)=n{c}_n{x}^{n-1}+(n-1)c_{n-1}{x}^{n-2}+(n-2)c_{n-2}{x}^{n-3}+\cdots +2c_{2}x+c_1$

Trigonometric Functions

$\frac{d}{dx}\sin (x)=\cos (x)$

$\frac{d}{dx}\cos (x)=-\sin (x)$

$\frac{d}{dx}\tan (x)={\sec }^2(x)$

$\frac{d}{dx}\cot (x)=-{\csc }^2(x)$

$\frac{d}{dx}\sec (x)=\sec (x)\tan (x)$

$\frac{d}{dx}\csc (x)=-\csc (x)\cot (x)$

Exponential and Logarithmic Functions

$\frac{d}{dx}{e}^x=e^x$

$\frac{d}{dx}{a}^x=a^x\ln (a)\text{if }a>0$

$\frac{d}{dx}\ln (x)=\frac{1}{x}$

$\frac{d}{dx}{\log }_a(x)=\frac{1}{\ln (a)x}\text{if }a>0,a\ne 1$

$\frac{d}{dx}(f^g)=\frac{d}{dx}\left(e^{g\ln (f)}\right)=f^g(\frac{df}{dx}\cdot \frac{g}{f}+\frac{dg}{dx}\cdot \ln (f)),f>0$

$\frac{d}{dx}(c^f)=\frac{d}{dx}\left(e^{f\ln (c)}\right)=\frac{df}{dx}\cdot c^f\ln (c)$

Inverse Trigonometric Functions

$\frac{d}{dx}\arcsin (x)=\frac{1}{\sqrt{1-x^2}}$

$\frac{d}{dx}\arccos (x)=-\frac{1}{\sqrt{1-x^2}}$

$\frac{d}{dx}\arctan (x)=\frac{1}{1+x^2}$

$\frac{d}{dx}\mathrm{arcsec}(x)=\frac{1}{|x|\sqrt{x^2-1}}$

$\frac{d}{dx}\mathrm{arccot}(x)=-\frac{1}{1+x^2}$

$\frac{d}{dx}\mathrm{arccsc}(x)=-\frac{1}{|x|\sqrt{x^2-1}}$

Hyperbolic and Inverse Hyperbolic Functions

$\frac{d}{dx}\sinh (x)=\cosh (x)$

$\frac{d}{dx}\cosh (x)=\sinh (x)$

$\frac{d}{dx}\tanh (x)={\mathrm{s}\mathrm{e}\mathrm{c}\mathrm{h}}^2(x)$

$\frac{d}{dx}\mathrm{s}\mathrm{e}\mathrm{c}\mathrm{h}(x)=-\tanh (x)\mathrm{s}\mathrm{e}\mathrm{c}\mathrm{h}(x)$

$\frac{d}{dx}\coth (x)=-{\mathrm{c}\mathrm{s}\mathrm{c}\mathrm{h}}^2(x)$

$\frac{d}{dx}\mathrm{c}\mathrm{s}\mathrm{c}\mathrm{h}(x)=-\coth (x)\mathrm{c}\mathrm{s}\mathrm{c}\mathrm{h}(x)$

$\frac{d}{dx}\mathrm{a}\mathrm{r}\mathrm{c}\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{h}(x)=\frac{1}{\sqrt{x^2+1}}$

$\frac{d}{dx}\mathrm{a}\mathrm{r}\mathrm{c}\mathrm{c}\mathrm{o}\mathrm{s}\mathrm{h}(x)=-\frac{1}{\sqrt{x^2-1}}$

$\frac{d}{dx}\mathrm{a}\mathrm{r}\mathrm{c}\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{h}(x)=\frac{1}{1-x^2}$

$\frac{d}{dx}\mathrm{a}\mathrm{r}\mathrm{c}\mathrm{s}\mathrm{e}\mathrm{c}\mathrm{h}(x)=\frac{1}{x\sqrt{1-x^2}}$

$\frac{d}{dx}\mathrm{a}\mathrm{r}\mathrm{c}\mathrm{c}\mathrm{o}\mathrm{t}\mathrm{h}(x)=-\frac{1}{1-x^2}$

$\frac{d}{dx}\mathrm{a}\mathrm{r}\mathrm{c}\mathrm{c}\mathrm{s}\mathrm{c}\mathrm{h}(x)=-\frac{1}{|x|\sqrt{1+x^2}}$

Reference

1. Derivative
2. Table_of_derivatives