# Differentiation of Fractions

## Derivative Calculator

Function of x

First derivative of the function after x

Input area for the fraction:

f(x) =

cl
$\frac{d}{dx}f\left(x\right)$
$\frac{{d}^{n}}{d{x}^{n}}f\left(x\right)$
Plot
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End
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μ
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${e}^{ax}$
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sin
cos
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ex
ln
xa
$\frac{a}{x}$
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asin
acos
atan
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ax
$\frac{a}{x+b}$
|x|
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sinh
cosh
$\frac{ax+c}{bx+d}$
$\frac{a+x}{b+x}$
$\frac{1+x}{1-x}$
$\frac{1}{a+bx}$
$\frac{1+\sqrt{x}}{1-\sqrt{x}}$
$\frac{{x}^{2}-{a}^{2}}{{x}^{2}+{a}^{2}}$
$\frac{\mathrm{sin}\left(x\right)}{\mathrm{cos}\left(x\right)}$
$\frac{x}{{e}^{x}}$
FunctionDescription
sin(x)Sine of x
cos(x)Cosine of x
tan(x)Tangent of x
asin(x)arcsine
acos(x)arccosine of x
atan(x)arctangent of x
atan2(y, x)Returns the arctangent of the quotient of its arguments.
cosh(x)Hyperbolic cosine of x
sinh(x)Hyperbolic sine of x
pow(a, b)Power ab
sqrt(x)Square root of x
exp(x)e-function
log(x), ln(x)Natural logarithm
log(x, b)Logarithm to base b
log2(x), lb(x)Logarithm to base 2
log10(x), ld(x)Logarithm to base 10
more ...

## Notations

Notation for Derivatives

$\frac{d}{dx}f\left(x\right)=\frac{df}{dx}\left(x\right)=\frac{df\left(x\right)}{dx}={f}^{\prime }\left(x\right)$

## Quotient Rule

The quotient rule specifies how to treat the quotient of two functions when differentiate.

Derivative rule for fractions:

$\frac{d}{dx}f\left(x\right)=\frac{d}{dx}\frac{u\left(x\right)}{v\left(x\right)}=\frac{v\left(x\right)\frac{d}{dx}u\left(x\right)-u\left(x\right)\frac{d}{dx}v\left(x\right)}{{v}^{2}\left(x\right)}=\frac{{u}^{\prime }\left(x\right)\cdot v\left(x\right)-u\left(x\right)\cdot {v}^{\prime }\left(x\right)}{{v}^{{}^{2}}}$

### Example of the application of the quotient rule

Example for the derivative of a fraction:

${\left(\frac{x+a}{x+b}\right)}^{\prime }$

Application of the quotient rule with u=x+a and v=x+b

$=\frac{{\left(x+a\right)}^{\prime }\left(x+b\right)-\left(x+a\right){\left(x+b\right)}^{\prime }}{{\left(x+b\right)}^{2}}$

Derivative of the terms results u′=1 and v′=1

$=\frac{x+b-\left(x+a\right)}{{\left(x+b\right)}^{2}}$

After simplification

$=\frac{b-a}{{\left(x+b\right)}^{2}}$

## More Calculators

Here is a list of of further useful calculators:

Index Derivative calculus Partial derivatives and gradient Derivative roots Derivative e-function Derivative sine cosine tangent Derivative sinh cosh tanh Derivative table Gradient calculator Gradient 2d Plot Function Plot ODE first order