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# Derivative calculator

##### Online calculator which determines first derivative of a function
Anton2011-07-29 16:49:33

This calculator finds derivative of entered function and tries to simplify the formula.

Use "Function" field to enter mathematical expression with x variable. You can use operations like addition +, subtraction -, division /, multiplication *, power ^, and common mathematical functions. Full syntax description can be found below the calculator.

Simplification of derivative formula can take great amount of time, especially for complex expressions. You can use "Stop" button to stop simplification and see current results. Usually 10-15 seconds yield good enough result.

##### Derivative calculator
Function derivative:

## Function formula syntax

In function notation you can use one variable (always use x), brackets, pi number (pi), exponent (e), operations: addition +, subtraction -, division /, multiplication *, power ^.
You can use following common functions: sqrt - square root,exp - power of exponent,lb - logarithm to base 2,lg - logarithm to base 10,ln - logarithm to base e,sin - sine,cos - cosine,tg - tangent,ctg - cotangent,sec - secant,cosec - cosecant,arcsin - arcsine,arccos - arccosine,arctg - arctangent,arcctg - arccotangent,arcsec - arcsecant,arccosec - arccosecant,versin - versien,vercos - vercosine,haversin - haversine,exsec - exsecant,excsc - excosecant,sh - hyperbolic sine,ch - hyperbolic cosine,th - hyperbolic tangent,cth - hyperbolic cotangent,sech - hyperbolic secant,csch - hyperbolic cosecant, abs - module, sgn - signum (sign), logP - logarithm to base P, f.e. log7(x) - logarithm to base 7, _rootP - P-th root, f.e. root3(x) - cubic root

## Finding derivative

To get derivative is easy using differentiation rules and derivatives of elementary functions table. The challenging task is to interpret entered expression and simplify the obtained derivative formula. I do my best to solve it, but it's another story.

### Differentiation rules

1) the sum rule:
$(u+v+...+w)'=u'+v'+...+w'$
2) the product rule:
$(uv)'=u'v+v'u$
3) the quotient rule:
$(\frac{u}{v})'=\frac{u'v-v'u}{v^2}$
4) the chain rule:
$y=f(u), u=\phi(x), y'=f'(u)\phi'(x)$

### Derivatives of common functions

The polynomial or elementary power:
$(x^{n})'=nx^{n-1}$
The exponential function:
$(a^{x})'=a^{x}\ln(a)$
$(e^{x})'=e^{x}$
The logarithmic function:
$(\ln(x))'=\frac{1}{x}$
The trigonometric functions:
$(\sin{x})'=\cos{x}$,
$(\cos(x))'=-\sin(x)$,
$(\tan(x))'=\frac{1}{\cos^2(x)}$,
$(\cot(x))'=-\frac{1}{\sin^2(x)}$
The inverse trigonometric functions:
$(\arcsin(x))'=\frac{1}{\sqrt{1-x^2}}$,
$(\arccos(x))'=-\frac{1}{\sqrt{1-x^2}}$,
$(arctg(x))'=\frac{1}{1+x^2}$,
$(arcctg(x))'=-\frac{1}{1+x^2}$
The hyperbolic functions:
$(sh(x))' = ch(x)$
$(ch(x))' = sh(x)$
$(th(x))' = -th(x)sech(x)$
$(cth(x))' = -csch^2(x)$

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