# Derivative Calculator

## Partial Derivative Calculator

The derivative calculator calculates the derivative or partial derivative of a function f. Additional the calculator computes the gradient in 3d.

Input field for the function to be derived. With 'ok' the entered function is accepted. With ∂/∂... the corresponding derivatives can be formed. Multiple application leads in each case to the derivative of the predecessor function.

f(...) =

cl
ok
dn / dxn
n / ∂xn
n / ∂yn
n / ∂zn
Pos1
End
7
8
9
/
Δ
x
y
z
4
5
6
*
Ω
a
b
c
1
2
3
-
μ
π
(
)
0
.
+
ω
sin
cos
tan
ex
ln
xa
a / x
^
σ
asin
acos
atan
x2
x
ax
a / x+b
|x|
δ
sinh
cosh
a⋅x+c / b⋅y+c
a+x / b+z
z2-a2/ z2+a2
a / x+b
1+√y / 1-√y
ea⋅xsin(y)cos(z)
x+a
ea⋅x

### Usable expressions in the definition of the function

Constants

NameDescription
LN2Natural logarithm of 2
LN10Natural logarithm of 10
LOG2EBase 2 logarithm of EULER
LOG10EBase 10 logarithm of EULER
PIRatio of the circumference of a circle to its diameter
SQRT1_2Square root of 1/2
SQRT2Square root of 2

Trigonometric Functions

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

Logarithm and Exponential

FunctionDescription
pow(b, e)e to the b
sqrt(x)Square root of x
exp(x)EULER to the x
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 functions

FunctionDescription
ceil(x)Get smallest integer n with n > x.
abs(x)Absolute value of x
max(a, b, c, ...)Maximum value of all given values.
min(a, b, c, ...)Minimum value of all given values.
random(max = 1)Generate a random number between 0 and max.
round(v)Returns the value of a number rounded to the nearest integer.
floor(x)Returns the biggest integer n with n < x.
factorial(n)Calculates n!
trunc(v, p = 0)Truncate v after the p-th decimal.
V(s)Returns the value of the given element, e.g. sliders and angles.

### Derivation rules in short

Factor rule: A constant factor is preserved when differentiate

$\left(a\cdot f\right)\prime =a\cdot f\prime$

Sum rule: When deriving a sum, the summands can be derived individually

$\left({f}_{1}+{f}_{2}\right)\prime ={f}_{1}\prime +{f}_{2}\prime$

Product rule: Rule for deriving products

$\left(u\cdot v\right)\prime =u\prime \cdot v+u\cdot v\prime$

Quotient rule: Rule for deriving quotients

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

Chain rule: Nested functions go into a product of the inner and outer derivatives when differentiated

$\left(f\left(g\left(x\right)\right)\right)\prime =f\prime \left(g\right)\cdot g\prime \left(x\right)$

Basic derivatives:

$\frac{d}{dx}\mathrm{Const.}=0$

$\frac{d}{dx}x=1$

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

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:

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

Derivation of trigonometric functions:

$\frac{d}{dx}\mathrm{sin}\left(x\right)=\mathrm{cos}\left(x\right)$

$\frac{d}{dx}\mathrm{cos}\left(x\right)=-\mathrm{sin}\left(x\right)$

$\frac{d}{dx}\mathrm{sin}\left(kx\right)$$=k\mathrm{cos}\left(kx\right)$

$\frac{d}{dx}\mathrm{cos}\left(kx\right)$$=-k\mathrm{sin}\left(kx\right)$

$\frac{d}{dx}\mathrm{tan}\left(x\right)$$=\frac{d}{dx}\frac{\mathrm{sin}\left(x\right)}{\mathrm{cos}\left(x\right)}$$=\frac{1}{{\mathrm{cos}}^{2}\left(x\right)}$

Derivations of the e-function:

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

$\frac{d}{dx}{e}^{ax}$$=\left({e}^{ax}\right)\prime$$=a{e}^{ax}$

$\frac{d}{dx}{e}^{a{x}^{2}}$$=\left({e}^{a{x}^{2}}\right)\prime$$=2ax{e}^{a{x}^{2}}$

$\frac{d}{dx}\frac{1}{{e}^{x}}$$=\left(\frac{1}{{e}^{x}}\right)\prime$$=\left({e}^{-x}\right)\prime$$=-{e}^{-x}$$=-\frac{1}{{e}^{x}}$

$\frac{d}{dx}{e}^{\mathrm{ln}\left(x\right)}$$=\left({e}^{\mathrm{ln}\left(x\right)}\right)\prime$$=\left(x\right)\prime$$=1$

$\frac{d}{dx}{e}^{{x}^{n}}$$=\left({e}^{{x}^{n}}\right)\prime$$=n{x}^{n-1}{e}^{{x}^{n}}$

$\frac{d}{dx}{\left({e}^{x}\right)}^{n}$$=\left({\left({e}^{x}\right)}^{n}\right)\prime$$=\left({e}^{nx}\right)\prime$$=n{e}^{nx}$

Derivation of the logarithm functions:

$\frac{d}{dx}\mathrm{ln}\left(x\right)=\frac{1}{x}$

$\frac{d}{dx}{\mathrm{log}}_{a}\left(x\right)=\frac{1}{x}{\mathrm{log}}_{a}\left(e\right)$

## Partial Derivatives

For functions with more than one variables the derivative to one of the variables is called partial derivative.

For a function with the variable x and several further variables the partial derivative to x is noted as follows.

$\frac{\mathrm{\partial }}{\mathrm{\partial }x}f\left(x,y,...\right)$

For partial derivation, the other variables are treated as constants.

## More Calculators

Here is a list of of further useful calculators:

Index Derivative calculus Derivative fraction 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