A partial derivative calculator is an online tool that computes the partial derivatives ∂f/∂x and ∂f/∂y of a two-variable function f(x,y), evaluates them at a chosen point, and visualizes the surface and tangent plane so learners can see how the derivatives describe local slope and direction.

How to use the Partial Derivative Calculator — step-by-step guide

Understanding partial derivatives is a foundational skill in multivariable calculus, applied mathematics, physics, engineering and machine learning. This partial derivative calculator was designed to make that learning intuitive: it computes symbolic partial derivatives, evaluates them numerically at any (x,y) point you choose, and displays a 3D surface plus the linear tangent plane using Plotly.js so you can visually confirm what the derivatives represent.

What the calculator does and why it’s useful

At its core the calculator performs three tasks:

1. Symbolic differentiation — it uses a JavaScript math engine to symbolically compute ∂f/∂x and ∂f/∂y so you can see exact expressions.
2. Numeric evaluation — it evaluates those derivatives at a specified point (x,y) and displays the numeric partials and the gradient vector ∇f.
3. Interactive visualization — using Plotly.js, it renders the surface z = f(x,y), highlights the selected point, and draws the tangent plane (linear approximation). This combination is especially helpful for learners who need to link algebraic manipulation to geometric intuition.

Why this matters: partial derivatives tell you how the surface changes when you vary one variable and hold the other fixed. They form the gradient, which points in the direction of steepest increase. Being able to compute and visualize these components accelerates comprehension and helps prevent common mistakes.

Step-by-step: Using the tool on your WordPress page

1 — Enter the function

Type your two-variable function in the f(x,y) field. The tool accepts common mathematical functions and operators: addition, subtraction, multiplication, division, exponentiation (^), and standard functions like sin(x), cos(y), exp(...), log(...), etc. Example inputs:

• x^2 + 3*x*y + sin(y)
• exp(x)*cos(y) - x*y^2
• log(x^2 + y^2 + 1)

Make sure to use valid syntax (for instance, use ^ for powers and sin(y) for sine).

2 — Choose the evaluation point (x, y)

Enter the coordinates where you want the partial derivatives evaluated. The calculator will compute:

• ∂f/∂x symbolically and evaluate it at (x,y)
• ∂f/∂y symbolically and evaluate it at (x,y)
  The results are displayed both as symbolic expressions (so you can check algebraic form) and as decimal approximations for immediate numerical insight.

3 — Pick a plot range

Select the plot range (for example ±5) to control how wide the surface grid is. A smaller range zooms in on local behavior; a larger range gives broader context.

4 — Compute and explore the Plotly visualization

Click Compute Partial Derivatives. The results panel will show the symbolic partial derivatives and their numeric values at the chosen (x,y). Below, the interactive Plotly 3D plot shows:

• The surface z = f(x,y) (rotatable and zoomable).
• A marker at the chosen point (x,y,f(x,y)).
• A translucent tangent plane built from the computed partials (visual linear approximation).

Plotly.js enables rotation, zooming, and hover interaction, which helps you explore how the tangent plane approximates the surface near the point.

Practical tips for learning and verification

• Check the symbolic result: If your computed symbolic partial seems off, re-check the function input for parentheses or operator mistakes.
• Use small ranges for local approximation: The tangent plane approximates the surface best in a small neighborhood — use ±1 or ±2 to inspect local linear behavior.
• Test simple examples: Try polynomials like x^2 + y^2 where ∂f/∂x = 2x and ∂f/∂y = 2y to validate the tool quickly.
• Explore directional meaning: The gradient ∇f = (∂f/∂x, ∂f/∂y) points to the steepest increase; compare gradient direction to the surface slope visually.

Technical notes (brief)

This calculator uses a JavaScript math library to parse and symbolically differentiate the input expression, and Plotly.js for plotting. It runs client-side, so no server round-trips are required; this keeps the tool fast and private. The code is intentionally responsive and sized (max-width 800px) so it fits neatly between sidebars in most WordPress themes when embedded as custom HTML.

Example workflows

• Learning: Input f(x,y) = x^2 + 3xy + sin(y), evaluate at (1, 0.5). See how ∂f/∂x and ∂f/∂y capture slopes in the x and y directions and how the tangent plane approximates near (1, 0.5).
• Homework checking: Compute your hand-worked derivatives symbolically and compare to the calculator’s output to quickly confirm correctness.
• Visualization: For functions with interesting curvature (e.g., saddles like f(x,y) = x^2 - y^2), use rotation to study how the tangent plane sits relative to the saddle.

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FAQ

Q: What types of functions can I enter?
A: Most elementary functions supported by math.js: polynomials, trig (sin, cos, tan), exponentials (exp), logarithms (log), roots, and compositions. Avoid undefined operations (e.g., log(negative number) at real-valued evaluation points).

Q: Are the symbolic derivatives exact?
A: The tool uses symbolic differentiation provided by the math library. For standard expressions it returns algebraically correct symbolic derivatives. Simplification follows library rules; when needed, manually simplify expressions for a cleaner algebraic form.

Q: What if the evaluator reports an error?
A: Check syntax for missing parentheses, unintended characters, or unsupported operations. The error message will indicate parsing or evaluation problems.

Q: Can I change plot resolution or style?
A: The embedded tool uses a practical default resolution for responsiveness. Developers can edit the code to increase grid resolution, change color scales, or adjust the tangent-plane opacity.

Q: Is user data sent to a server?
A: No. The calculator runs client-side in the browser, so function strings and point values are processed locally.