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Partial Derivative Calculator

Enter your multivariable function, and choose the variable for differentiation. After that, click on the 'Calculate' button to get the partial derivative instantly!

Enter a function f(x,y): Hint: Please write e^x Derivative e^{x}

as W.R.T:

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Partial Derivative Calculator

The online Partial Derivative Calculator differentiates multivariable functions step-by-step with respect to the chosen variable, treating all other variables as constants.

What is a Partial Derivative?

A partial derivative is the derivative of a multivariable function with respect to one of its variables while keeping the others constant.

It measures how the function changes as one variable changes. The symbol ∂/∂ denotes partial derivatives.

Understanding Partial Derivatives

For a function with two independent variables, f(x, y):

Partial derivative with respect to x: ∂f/∂x or fₓ (treat y as constant).
Partial derivative with respect to y: ∂f/∂y or fᵧ (treat x as constant).

Partial Derivative Formulas

With respect to x:

∂f/∂x = lim_h→0 [(f(x + h, y) - f(x, y)) / h]

With respect to y:

∂f/∂y = lim_h→0 [(f(x, y + h) - f(x, y)) / h]

How to Calculate a Partial Derivative

Identify the multivariable function f(x, y, z,...) and the variable to differentiate (x, y, z, etc.).
Keep all other variables constant.
Apply standard differentiation rules (power, product, chain, quotient) with respect to the chosen variable.

Note: Any term not containing the variable of differentiation is treated as a constant and its derivative is zero.

Second Order Partial Derivatives

fxx = ∂²f/∂x²: Derivative of ∂f/∂x with respect to x.
fyy = ∂²f/∂y²: Derivative of ∂f/∂y with respect to y.

Example

Find ∂f/∂x for f(x, y) = 2x² + eʸ - 3xy²:

Variable: x, treat y as constant.
Derivative of each term:
- 2x² → 4x
- eʸ → 0
- -3xy² → -3y²
Combine: ∂f/∂x = 4x - 3y²

Rules of Partial Derivatives

1. Product Rule

If u = f(x, y)·g(x, y):

uₓ = g·∂f/∂x + f·∂g/∂x

uᵧ = g·∂f/∂y + f·∂g/∂y

2. Quotient Rule

If u = f(x, y)/g(x, y), g ≠ 0:

uₓ = (g·∂f/∂x - f·∂g/∂x) / g²

uᵧ = (g·∂f/∂y - f·∂g/∂y) / g²

3. Power Rule

uₓ = n·[f(x, y)]ⁿ⁻¹ · ∂f/∂x

uᵧ = n·[f(x, y)]ⁿ⁻¹ · ∂f/∂y

4. Chain Rule

For dependent variables:

One independent variable: z = f(x(t), y(t))

∂z/∂t = ∂z/∂x · dx/dt + ∂z/∂y · dy/dt

Two independent variables: z = f(x(u,v), y(u,v))

∂z/∂u = ∂z/∂x · ∂x/∂u + ∂z/∂y · ∂y/∂u

∂z/∂v = ∂z/∂x · ∂x/∂v + ∂z/∂y · ∂y/∂v

Partial Derivative of Natural Logarithm (ln)

Apply chain, product, or quotient rules as required.
Treat other variables as constants.
Compute derivatives for each variable involved.

Using the Partial Derivative Calculator

Enter the multivariable function.
Select the variable for differentiation.
Click "Calculate" to get the partial derivative with step-by-step solutions.

Applications of Partial Derivatives

Physics: Maxwell’s equations, Schrödinger equation, thermodynamics.
Economics: Optimization of production, utility, and cost functions.
Computer Science: Optimization in machine learning and AI algorithms.
Medicine: Image reconstruction in MRI and CT scans.
Environmental Science: Modeling populations, pollution, and solving differential equations.

FAQs

Can this calculator handle multiple variables?

Yes, you can specify which variable to differentiate while treating others as constants.

Difference between regular derivative and partial derivative?

A regular derivative applies to single-variable functions. Partial derivatives apply to multivariable functions, differentiating with respect to one variable at a time while keeping others constant.

References

Wikipedia: Partial Derivative

Khan Academy: Introduction to Partial Derivatives

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