# 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 ${\displaystyle f(x)=y}$ . The rate of change of ${\displaystyle f(x)}$ correspond to change of ${\displaystyle x}$ is equal to the ratio of change in ${\displaystyle f(x)}$ over change in ${\displaystyle x}$

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

Then the Derivative of the function is defined as

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

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

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

## Table of Derivative

### General Rules

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

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

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

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

### Powers and Polynomials

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

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

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

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

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

${\displaystyle {{\frac {d}{dx}}(c_{n}x^{n}+c_{n-1}x^{n-1}+c_{n-2}x^{n-2}+\cdots +c_{2}x^{2}+c_{1}x+c_{0})=nc_{n}x^{n-1}+(n-1)c_{n-1}x^{n-2}+(n-2)c_{n-2}x^{n-3}+\cdots +2c_{2}x+c_{1}}}$

### Trigonometric Functions

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

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

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

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

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

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

### Exponential and Logarithmic Functions

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

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

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

${\displaystyle {\frac {d}{dx}}\log _{a}(x)={\frac {1}{\ln(a)x}}\qquad {\mbox{if }}a>0,a\neq 1}$

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

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

### Inverse Trigonometric Functions

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

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

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

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

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

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

### Hyperbolic and Inverse Hyperbolic Functions

${\displaystyle {\frac {d}{dx}}\sinh(x)=\cosh(x)}$
${\displaystyle {\frac {d}{dx}}\cosh(x)=\sinh(x)}$
${\displaystyle {\frac {d}{dx}}\tanh(x)={\rm {sech}}^{2}(x)}$
${\displaystyle {\frac {d}{dx}}{\rm {sech}}(x)=-\tanh(x){\rm {sech}}(x)}$
${\displaystyle {\frac {d}{dx}}\coth(x)=-{\rm {csch}}^{2}(x)}$
${\displaystyle {\frac {d}{dx}}{\rm {csch}}(x)=-\coth(x){\rm {csch}}(x)}$
${\displaystyle {\frac {d}{dx}}{\rm {arcsinh}}(x)={\frac {1}{\sqrt {x^{2}+1}}}}$
${\displaystyle {\frac {d}{dx}}{\rm {arccosh}}(x)=-{\frac {1}{\sqrt {x^{2}-1}}}}$
${\displaystyle {\frac {d}{dx}}{\rm {arctanh}}(x)={\frac {1}{1-x^{2}}}}$
${\displaystyle {\frac {d}{dx}}{\rm {arcsech}}(x)={\frac {1}{x{\sqrt {1-x^{2}}}}}}$
${\displaystyle {\frac {d}{dx}}{\rm {arccoth}}(x)=-{\frac {1}{1-x^{2}}}}$
${\displaystyle {\frac {d}{dx}}{\rm {arccsch}}(x)=-{\frac {1}{|x|{\sqrt {1+x^{2}}}}}}$