1. What is Multivariable Derivative?

Multivariable derivatives, also known as partial derivatives, measure how a function changes as one variable changes while keeping other variables constant. They are fundamental in multivariable calculus and have applications in physics, engineering, and economics.

2. How Does the Calculator Work?

The calculator computes partial derivatives of a function with respect to specified variables:

Where:

• \( f \) — Multivariable function
• \( x \) — First variable
• \( y \) — Second variable

Explanation: The partial derivative with respect to x measures the rate of change of f as x changes, holding y constant, and vice versa.

3. Importance of Partial Derivatives

Details: Partial derivatives are essential for optimization problems, gradient computation, tangent plane determination, and in various physical applications such as heat transfer and fluid dynamics.

4. Using the Calculator

Tips: Enter the function expression (e.g., "x^2 + y^2"), specify the variables (default: x and y). The calculator will compute the partial derivatives with respect to each variable.

5. Frequently Asked Questions (FAQ)

Q1: What types of functions can be processed?
A: The calculator can handle polynomial, trigonometric, exponential, and logarithmic functions of multiple variables.

Q2: How are mixed partial derivatives handled?
A: The calculator computes first-order partial derivatives. Higher-order and mixed derivatives require additional computation steps.

Q3: What notation is used for input?
A: Use standard mathematical notation: ^ for exponentiation, * for multiplication, sin(x), cos(x), exp(x), log(x), etc.

Q4: Are there limitations to this calculator?
A: The calculator may have limitations with extremely complex functions or implicit differentiation. For advanced computations, specialized mathematical software is recommended.

Q5: Can I compute derivatives with respect to more than two variables?
A: This calculator is designed for two variables. For functions with more variables, the process would need to be repeated for each additional variable.