Enter your multivariable function, and choose the variable for differentiation. After that, click on the 'Calculate' button to get the partial derivative instantly!
This partial derivative calculator differentiates the multivariable functions step-by-step with respect to your chosen variable and considers the other variables as constants.
"The partial derivative is defined as the derivative of a multivariable function with respect to one of its variables, keeping all the variables as constants."
It measures how a function changes when one of its variables changes, treating the other variables as constants.
The symbol "∂/∂" is used to indicate partial differentiation.
Consider a function with two independent variables, f(x, y):
To find the partial derivative of a function with respect to “x”:
∂f/∂x = limh→0 [ (f(x + h, y) - f(x, y)) / h ]
To find the partial derivative of a function with respect to “y”:
∂f/∂x = limh→0 [ (f(x, y + h) - f(x, y)) / h ]
Note: All terms that do not contain the chosen variable will be considered as a constant, and their derivative will be zero.
Second Partial Derivatives:
Second order partial derivatives are found by taking the partial derivative of a first-order partial derivative. For a function f(x,y):
Find partial derivative of the function f(x, y) = 2x² + eʸ - 3xy² with respect to x, denoted as ∂f/∂x.
Solution:
Step #1: Identify the function and the variable of differentiation
We are given the function f(x,y)= 2x²+eʸ−3xy² and we want to differentiate with respect to x.
Step #2: Consider y as a constant
This means that terms involving only y (like ey) will have a derivative of zero with respect to x.
Step #3: Apply standard differentiation rules
Using the power rule (dxd(xn)=nxn−1), the derivative of 2x² is 2×2x²⁻¹ = 4x¹ = 4x.
Since we are keeping y as a constant, the derivative of eʸ with respect to x is 0
Here, -3y² is treated as a constant coefficient. The derivative of x with respect to x is 1. So, the derivative of - 3xy² with respect to x is - 3y²×1 = −3y²
Step #4: Add the derivatives of each term
We get: ∂f/∂x = 4x + 0 - 3y²
Step #5: Simplify the result: ∂f/∂x = 4x - 3y²
Therefore, the partial derivative of f(x, y) = 2x²+eʸ−3xy² with respect to x is 4x - 3y².
To verify your work or to quickly compute partial derivatives for multivariable functions, use our multivariable partial derivative calculator. It provides detailed, step-by-step solutions to help you understand the process.
If u = f(x, y).g(x, y):
ux = ∂u/∂x = g(x, y) * ∂f/∂x + f(x, y) * ∂g/∂x
And, uy = ∂u/∂y = g(x, y) * ∂f/∂y + f(x, y) * ∂g/∂y
u = f(x, y)/g(x, y), where g(x,y) ≠ 0:
ux = (g(x, y) * ∂f/∂x - f(x, y) * ∂g/∂x) / [g(x, y)]2
And, uy = (g(x, y) * ∂f/∂y - f(x, y) * ∂g/∂y) / [g(x, y)]2
ux = n |f(x, y)|n-1 ∂f/∂x
And, uy = n |f(x, y)|n-1 ∂f/∂y
In the chain rule, each variable of a multivariable function is the function of another variable. Therefore, the derivative of the function is the sum of the chain rule, applied to all the variables.
Chain Rule for One Independent Variable:
If z = f(x(t), y(t)):
∂z/∂t = ∂z/∂x ⋅ ∂x/∂t + ∂z/∂y ⋅ ∂y/∂t
Chain Rule for Two Independent Variables:
If z = f(x(u,v), y(u,v)):
∂z/∂u = ∂z/∂x ⋅ ∂x/∂u + ∂z/∂y ⋅ ∂y/∂u
and
∂z/∂v = ∂z/∂x ⋅ ∂x/∂v + ∂z/∂y ⋅ ∂y/∂v
Follow these steps to correctly use our online partial derivative calculator to differentiate the given functions:
Partial derivatives have a great use in various fields of life, including:
Yes, our calculator can handle functions containing multiple variables. When you input your function, you can specify which variable you want to differentiate with respect to at that time.
The main difference between a regular derivative and a partial derivative lies in the function upon which they operate. A regular derivative differentiates functions that contain only one independent variable. In contrast, partial derivatives are used to differentiate multivariable functions, but the differentiation is performed with respect to one specific variable at a time, keeping all the other variables as constants.
References:
From the source of Wikipedia.org: Partial Derivative.
From the source of KhanAcademy.org: Introduction to Partial Derivatives.
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