Gradient Calculator

This Gradient Calculator helps you find the gradient vector of a multivariable function and evaluate it at a given point if needed. The calculator finds the required partial derivatives and combines them to form ∇f in two or three variables. If a point is provided, it then substitutes the coordinates to evaluate the gradient there. It is a simple way to check answers, understand the method clearly, and practise multivariable calculus step by step.

Step-by-step method

1. Identify the variables and the function f.
2. Compute each partial derivative with substeps.
3. Combine them into the gradient vector ∇f.
4. If a point is given, evaluate ∇f at the point.
5. Compute the magnitude of the gradient at the point.

Formulas:

Example 1: f(x,y)=x^2+y^2; (2,1)

Step 1 - Identify the variables and the function.

In this problem: Variables: x, y | Point provided.

Step 2A - Differentiate term 1 with respect to x.

In this problem: Rule shown by the math.

Step 2B - Differentiate term 2 with respect to x.

In this problem: Rule shown by the math.

Step 2C - Combine the results.

In this problem: Add the derivatives from the previous substeps.

Step 3A - Differentiate term 1 with respect to y.

In this problem: Rule shown by the math.

Step 3B - Differentiate term 2 with respect to y.

In this problem: Rule shown by the math.

Step 3C - Combine the results.

In this problem: Add the derivatives from the previous substeps.

Step 4 - Combine into the gradient vector.

In this problem: Put the partial derivatives into ⟨ … ⟩.

Step 5A - Substitute the point into fx.

In this problem: Substitute the point values.

Step 5B - Simplify fx at the point.

In this problem: Simplify the substituted expression.

Step 5C - Substitute the point into fy.

In this problem: Substitute the point values.

Step 5D - Simplify fy at the point.

In this problem: Simplify the substituted expression.

Step 6 - Gradient at the point.

In this problem: Put the evaluated components into ⟨ … ⟩.

Step 7A - Magnitude of the gradient.

In this problem: Use the magnitude formula.

Step 7B - Simplify the magnitude.

In this problem: Simplify the square root.

Final answer: Gradient computed