# Derivation of root functions

## Derivative Calculator

Function of x

First derivative of the function after x

Input field for the root function:

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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Δ
x
y
z
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Ω
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b
c
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μ
π
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0
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${e}^{ax}$
+
ω
sin
cos
tan
ex
ln
xa
$\frac{a}{x}$
^
σ
asin
acos
atan
x2
x
ax
$\frac{a}{x+b}$
|x|
δ
sinh
cosh
$\sqrt{x+a}$
$\sqrt[3]{x+a}$
$\sqrt[3]{x}$
$\sqrt[4]{x}$
$\frac{\sqrt{a+x}}{\sqrt{b-x}}$
${e}^{\sqrt{x}}$
$\frac{1}{\sqrt{ax}}$
$\sqrt{{e}^{ax}}$
$\sqrt{a{x}^{2}+bx+c}$
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

Notations for roots

$\sqrt[n]{x}={x}^{\frac{1}{n}}$

Notations for derivatives

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

## Derivation of root functions

Derivative n-th root

$\frac{d}{dx}\sqrt[n]{x}=\frac{d}{dx}{x}^{\frac{1}{n}}=\frac{1}{n}\cdot {x}^{\frac{1}{n}-1}=\frac{1}{n}\cdot {x}^{\frac{1-n}{n}}=\frac{1}{n}\cdot \sqrt[n]{{x}^{1-n}}=\frac{1}{n\cdot \sqrt[n]{{x}^{n-1}}}$

Derivation square root

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

Derivation cube root

$\frac{d}{dx}\sqrt[3]{x}=\frac{d}{dx}{x}^{\frac{1}{3}}=\frac{1}{3}\cdot {x}^{\frac{1}{3}-1}=\frac{1}{3\cdot \sqrt[3]{{x}^{2}}}$

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