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
grad(f) ∇f
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_2Quadratwurzel von 1/2
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

( af ) = af

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

( f1 + f2 ) = f1 + f2

Product rule: Rule for deriving products

( uv ) = uv + uv

Quotient rule: Rule for deriving quotients

( u v ) = uv-uv v2

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

( f(g(x)) ) = f(g)g(x)

Basic derivatives:

d d x Const. = 0

d d x x = 1

d d x xn = nxn-1

Derivative n-th root:

d d x xn = d d x x1n = 1nx1n-1 = 1nx1-nn = 1nx1-nn = 1nxn-1n

Derivation square root:

d d x x = 12x

Derivation cube root:

d d x x3 = d d x x13 = 13x13-1 = 13x23

Derivation of trigonometric functions:

d d x sin(x) = cos(x)

d d x cos(x) = -sin(x)

d d x sin(kx) = kcos(kx)

d d x cos(kx) = -ksin(kx)

d d x tan(x) = d d x sin(x) cos(x) = 1 cos2(x)

Derivations of the e-function:

d d x ex = (ex) = ex

d d x eax = (eax) = aeax

d d x eax2 = (eax2) = 2axeax2

d d x 1ex = (1ex) = (e-x) = -e-x = -1ex

d d x eln(x) = (eln(x)) = (x) = 1

d d x exn = (exn) = nxn-1exn

d d x (ex)n = ((ex)n) = (enx) = nenx

Derivation of the logarithm functions:

d d x ln(x) = 1x

d d x loga(x) = 1xloga(e)

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.

x f ( x , y , . . . )

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