Derivation of root functions

Derivative Calculator

Function of x

First derivative of the function after x

Input field for the root function:

f(x) =

\frac{d}{d x} f (x)

\frac{d^{n}}{d x^{n}} f (x)

Plot

Pos1

End

(

)

e^{a x}

sin

cos

tan

e^x

x^a

\frac{a}{x}

asin

acos

atan

x²

√x

a^x

\frac{a}{x + b}

|x|

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{a x}}

\sqrt{e^{a x}}

\sqrt{a x^{2} + b x + c}

Function	Description
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 a^b
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

$\frac{d}{d x} f (x) = \frac{d f}{d x} (x) = \frac{d f (x)}{d x} = f^{'} (x)$

Derivation of root functions

Derivative n-th root

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

Derivation square root

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

Derivation cube root

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

More Calculators

Here is a list of of further useful calculators:

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