Polynomial gradient
Find the gradient of x^2+y^2
Method: Differentiate with respect to each variable separately.
- 1Holding y constant gives f_x=2x.
- 2Holding x constant gives f_y=2y.
- 3Place both partials in a vector.
Final answer
Free online calculator
Gradient Calculator
Compute the gradient vector of any scalar function of two or three variables. Get each partial derivative with step-by-step work.
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How to Use the Gradient Calculator
1
Enter the function
Type f(x, y) or f(x, y, z) using standard notation.
2
Compute partials
We differentiate with respect to each variable in turn.
3
Form the gradient
The partial derivatives combine into the gradient vector.
Example
How the Gradient Calculator Chooses a Method
Compute a gradient vector and understand how each partial derivative contributes to direction of change.
The gradient is assembled by differentiating once with respect to each independent variable while holding the others constant. At a point, the vector points in the direction of steepest local increase.
Supported problem types
- Two-variable and common multivariable scalar functions
- Symbolic first partial derivatives
- Gradient evaluation at specified points
Limits and assumptions
- The gradient applies to scalar-valued functions, not directly to vector fields.
- A zero gradient marks a critical point but does not by itself classify it.
Worked Gradient Calculator Examples
Each example names the method, shows the ordered reasoning, and keeps the final answer separate so you can check your own work.
Mixed term
Find the gradient of xy+y^2
Method: Treat the other variable as constant in each partial.
- 1With respect to x, xy differentiates to y.
- 2With respect to y, xy+y^2 differentiates to x+2y.
- 3Assemble the ordered components.
Final answer
Evaluate at a point
Evaluate grad(x^2+y^2) at (1,-2)
Method: Find the symbolic gradient before substituting.
- 1The gradient is <2x,2y>.
- 2Substitute x=1 and y=-2.
- 3Evaluate each component.
Final answer
Learn This Topic
What the Gradient Calculator Helps You Understand
Compute the gradient vector of any scalar function of two or three variables. Get each partial derivative with step-by-step work.
Learning focus 1
Type f(x, y) or f(x, y, z) using standard notation.
Learning focus 2
We differentiate with respect to each variable in turn.
Learning focus 3
The partial derivatives combine into the gradient vector.
Input tips
- • Type f(x, y) or f(x, y, z) using standard notation.
- • The calculator supports functions of two or three variables. For more variables, enter the expression and we compute each partial.
Common mistakes to avoid
- • Differentiating every variable instead of holding the others constant.
- • Returning a scalar when the requested gradient is a vector.
Frequently Asked Questions
How many variables can the function have?
The calculator supports functions of two or three variables. For more variables, enter the expression and we compute each partial.
Does it show the direction of steepest ascent?
Yes. The gradient vector points in the direction of steepest increase of the function at each point.
Why is a gradient a vector?
It stores one partial derivative for each input direction, so both magnitude and direction matter.
What does a zero gradient mean?
It identifies a critical point; additional tests are needed to determine whether it is a minimum, maximum, or saddle point.