Derivation of the e-function

Derivative Calculator

Function of x

First derivative of the function after x

Input field for the e-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

e^{\sqrt{x}}

a e^{- b x^{2} + c}

\sqrt{e^{a x}}

a e^{b x + c}

e^{a x^{2}}

\frac{1}{e^{a x}}

\frac{x}{e^{x}}

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 derivatives:

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

Notations of the natural exponential function:

$e^{x} = \exp (x)$

The basis is the Euler number:

$e = 2.71828182845904523536028747135266249...$

Derivatives of the e-function

The e-function is the solution of the differential equation

$y' = y$

The chain rule for the e-function

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

Derivations of the e-function

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

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

$\frac{d}{d x} e^{a x^{2}}$ $= (e^{a x^{2}})'$ $= 2 a x e^{a x^{2}}$

$\frac{d}{d x} \frac{1}{e^{x}}$ $= (\frac{1}{e^{x}})'$ $= (e^{- x})'$ $= - e^{- x}$ $= - \frac{1}{e^{x}}$

$\frac{d}{d x} e^{\ln (x)}$ $= (e^{\ln (x)})'$ $= (x)'$ $= 1$

$\frac{d}{d x} e^{x^{n}}$ $= (e^{x^{n}})'$ $= n x^{n - 1} e^{x^{n}}$

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

Derivation of the logarithm functions

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

$\frac{d}{d x} \log_{a} (x) = \frac{1}{x} \log_{a} (e)$

More Calculators

Here is a list of of further useful calculators:

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