1. What is Multivariable Gradient?

The gradient of a multivariable function is a vector that points in the direction of the greatest rate of increase of the function. For a function f(x,y), the gradient is denoted as ∇f and contains all its partial derivatives.

2. How Does the Calculator Work?

The calculator computes the gradient using the formula:

Where:

• \( \nabla f \) — Gradient vector
• \( \frac{\partial f}{\partial x} \) — Partial derivative with respect to x
• \( \frac{\partial f}{\partial y} \) — Partial derivative with respect to y

Explanation: The gradient represents the direction and magnitude of steepest ascent of the function at any given point.

3. Importance of Gradient Calculation

Details: Gradient calculation is fundamental in optimization, machine learning, physics, and engineering. It's used in gradient descent algorithms, vector calculus, and analyzing multivariable functions.

4. Using the Calculator

Tips: Enter your multivariable function (e.g., "x^2 + y^2", "sin(x)*cos(y)"), specify the variables (default: x and y), and click calculate to see the gradient and partial derivatives.

5. Frequently Asked Questions (FAQ)

Q1: What is the geometric interpretation of gradient?
A: The gradient points in the direction of steepest ascent of the function, and its magnitude indicates the rate of increase in that direction.

Q2: Can this calculator handle more than 2 variables?
A: This version handles 2 variables (x,y). For more variables, the gradient extends naturally: ∇f = (∂f/∂x, ∂f/∂y, ∂f/∂z, ...)

Q3: What functions can I input?
A: You can input polynomial, trigonometric, exponential, and logarithmic functions. Use standard mathematical notation.

Q4: How is this different from directional derivative?
A: The gradient is a vector, while directional derivative is a scalar representing the rate of change in a specific direction.

Q5: What if my function has constraints?
A: This calculator computes the standard gradient. For constrained optimization, you would need Lagrange multipliers.