Calculator
Enter the first partial derivative expressions and the point where you want the gradient evaluated. Use z and ∂f/∂z as 0 for a two-variable case.
Supported functions include sin, cos, tan, asin, acos, atan, sqrt, abs, log, exp, sinh, cosh, tanh, floor, ceil, round, pow, pi, and e.
Example Data Table
| ∂f/∂x | ∂f/∂y | ∂f/∂z | Point (x, y, z) | Gradient | Unit Steepest Descent |
|---|---|---|---|---|---|
| 2x + y | x + 4y | 6z | (1, 2, 1) | (4, 9, 6) | (-0.3468, -0.7804, -0.5203) |
| 3x^2 | 2y | 0 | (2, -1, 0) | (12, -2, 0) | (-0.9864, 0.1644, 0) |
| y + z | x + z | x + y | (1, 3, 2) | (5, 3, 4) | (-0.7071, -0.4243, -0.5657) |
Formula Used
The negative gradient points toward the fastest local decrease of the function. When the gradient is zero, the point is stationary and no single steepest descent direction is determined from first-order information alone.
How to Use This Calculator
- Enter the first partial derivative with respect to x.
- Enter the first partial derivative with respect to y.
- Enter the first partial derivative with respect to z, or use 0 for two variables.
- Type the point coordinates where the derivatives should be evaluated.
- Optionally enter a custom direction vector to compare its directional derivative.
- Choose the number of decimal places you want displayed.
- Press Calculate to show the result below the header and above the form.
- Use the CSV or PDF buttons to export the computed summary.
Why This Calculator Helps
This tool evaluates user-supplied first partial derivatives at a chosen point and returns the gradient, the negative gradient, the normalized descent direction, and the directional derivative bounds. It is useful for multivariable calculus, optimization methods, numerical analysis, and classroom verification.
Because steepest descent depends on the local gradient, the calculator focuses on pointwise behavior. That makes it practical for checking hand calculations, preparing reports, exploring stationary points, and studying how search directions are selected in iterative minimization methods.
FAQs
1. What is the direction of steepest descent?
The direction of steepest descent is the unit vector opposite the gradient. It points toward the fastest local decrease of the function at the chosen point.
2. Why does the calculator need partial derivatives?
Steepest descent comes directly from the gradient, and the gradient is built from first partial derivatives. Without them, the calculator cannot determine the descent direction at the point.
3. What happens if the gradient is zero?
A zero gradient means the point is stationary. In that case, first-order information alone does not give one unique steepest descent direction.
4. Can I use this for two-variable functions?
Yes. Enter your x and y partial derivatives normally, and use 0 for the z partial derivative and z-coordinate.
5. What is the difference between negative gradient and unit descent direction?
The negative gradient gives the correct descent vector but keeps the original magnitude. The unit descent direction has length one and is better when you want only direction.
6. What does the custom direction option do?
It normalizes your entered vector and computes the directional derivative along that direction. This helps compare your chosen search direction with the true steepest descent direction.
7. Why are maximum and minimum directional derivatives shown?
They summarize the largest possible local increase and decrease at the point. Their magnitudes equal the gradient magnitude, with opposite signs.
8. Is this useful for optimization methods?
Yes. Gradient descent and related methods repeatedly move along descent directions. This calculator helps verify those directions at specific points before choosing a step size.