Calculate partial derivatives of multivariable functions with respect to x, y, or z. Use polynomial expressions, parentheses, powers, trigonometric functions, logarithms, exponentials, square roots, and more. Calculate one derivative, all first partial derivatives, second or mixed partial derivatives, and evaluate derivatives at a point.
x^2*y + y^3
,
choose
x,
and calculate.
The result is
2xy
because y is treated as a constant when differentiating with respect to x.
A partial derivative describes how a function of several variables changes when one variable changes while the other variables are held constant. For example, for f(x,y) , the two first partial derivatives are ∂f/∂x and ∂f/∂y .
For a function of three variables, f(x,y,z) , there are three first partial derivatives:
For \[ f(x,y)=x^2y+y^3 \]
\[ \frac{\partial f}{\partial x} = 2xy \]
For \[ f(x,y)=3x^2+4xy+y^2 \]
\[ \frac{\partial f}{\partial y} = 4x+2y \]
For \[ f(x,y,z)=xyz \]
\[ \frac{\partial f}{\partial z} = xy \]
For \[ f(x,y)=\sin(xy) \]
\[ \frac{\partial f}{\partial x} = y\cos(xy) \]
The calculator uses a symbolic parser rather than executing the submitted expression as PHP code.
| Category | Examples |
|---|---|
| Variables |
x,
y,
z
|
| Arithmetic |
+,
-,
*,
/
|
| Implicit multiplication |
2x,
xy,
xyz,
2sin(x)
|
| Powers |
x^2,
x^-2,
(x+y)^2
|
| Trigonometric |
sin,
cos,
tan,
cot,
sec,
csc
|
| Inverse trigonometric |
asin,
acos,
atan
|
| Hyperbolic |
sinh,
cosh,
tanh
|
| Logarithms and exponential |
ln(x),
log(x),
exp(x),
e^x
|
| Roots and absolute values |
sqrt(x),
abs(x)
|
| Constants |
pi,
e
|
A second partial derivative differentiates a function twice. Examples include:
A mixed partial derivative differentiates with respect to two different variables. For sufficiently smooth functions, mixed partial derivatives often agree:
The gradient collects all first partial derivatives into a vector. For f(x,y,z) :
For a dedicated gradient calculation, use the Gradient Calculator .
A partial derivative measures how a multivariable function changes with respect to one variable while the other variables are held constant.
∂f/∂x is the partial derivative of f with respect to x. While differentiating with respect to x, y and z are treated as constants.
An ordinary derivative is normally used for a function of one variable. A partial derivative is used for a function of several variables and describes the change with respect to one selected variable.
Yes. Select All first partial derivatives to calculate ∂f/∂x, ∂f/∂y, and ∂f/∂z.
The calculator supports x, y, z, arithmetic operations, powers, parentheses, implicit multiplication, trigonometric functions, inverse trigonometric functions, hyperbolic functions, logarithms, exponential functions, square roots, and absolute values.
Yes. Enable point evaluation and enter numerical values for the variables used by the expression.
The gradient is the vector of first partial derivatives. For f(x,y,z), the gradient is (∂f/∂x, ∂f/∂y, ∂f/∂z).
A mixed partial derivative differentiates with respect to two variables, such as ∂²f/(∂x∂y).