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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 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?

Step #1: Determine the function f(x,y,z,...) and the variable you want to differentiate with respect to (e.g., x, y, z)
Step #2: Keep all the other variables as constants. Treat them as fixed numbers while performing the differentiation process
Step #3: Now, implement the standard rules of differentiation (power rule, product rule, chain rule, etc.) according to the selected variable

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):

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

What Are the 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.

How to Use the Partial Derivative Calculator?

Follow these steps to correctly use our online partial derivative calculator to differentiate the given functions:

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

FAQ's:

Can this Calculator Handle Multiple Variables?

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

What Is The Difference Between a Regular Derivative and a Partial Derivative?

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:

Wikipedia: Partial Derivative.
Khan Academy: Introduction to Partial Derivatives.

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