Partial Derivative Calculator
Find the partial derivative of a multivariable function f(x, y, z, t) with respect to one variable — first, second, or third order — with an optional numeric evaluation at a point and a full step-by-step derivation.
Input
Use whichever of x, y, z, t your function needs — the rest can be left out.
Optional — only used if the derivative depends on x.
Optional.
Optional.
Optional.
Output
More ways to use this tool
REST API
curl -X POST https://api.iotools.cloud/v1/tool/partial-derivative-calculator \
-H "Authorization: Bearer YOUR_API_KEY" \
-H "Content-Type: application/json" \
-d '{
"f": "x^2*y + sin(x*y)",
"variable": "x",
"order": "1"
}'Swap in your own key from your account. The tool's fields are the body — no wrapper.
Ask an AI agent
Use the IOTools `partial-derivative-calculator` tool (Partial Derivative Calculator) on this input:
YOUR_INPUT_HEREPaste this at any agent connected to the IOTools MCP server, then add your input.
Embed widget
<iframe
src="https://iotools.cloud/embed/partial-derivative-calculator/"
width="100%" height="520" frameborder="0" scrolling="no" loading="lazy"
title="Partial Derivative Calculator — iotools.cloud"
sandbox="allow-scripts allow-forms allow-same-origin allow-downloads allow-popups allow-popups-to-escape-sandbox"
allow="clipboard-write"
style="width:100%;border:1px solid #e5e7eb;border-radius:12px;overflow:hidden"></iframe>
<script src="https://iotools.cloud/embed.js" async></script>Drop this into your own page — free, no key required, just a link back.
| Cost per API/MCP call | From 5 credits |
|---|---|
| Need more credits? | View pricing |
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Guides
The Partial Derivative Calculator finds the partial derivative of a multivariable function f(x, y, z, t) with respect to one variable you choose, treating every other variable as a constant. It supports the 1st, 2nd, and 3rd derivative with respect to that same variable, can optionally evaluate the result at a numeric point, and shows a full step-by-step derivation.
It's built for multivariable calculus students checking a homework answer, and for anyone who needs a partial derivative without differentiating by hand and tracking which variables are held constant.
How to use it
- Enter your function f(x, y, z, t) — standard math notation works:
^for powers,*for multiplication, plussqrt(),sin(),cos(),log(), and the other functions supported by mathjs's expression parser. Use whichever of x, y, z, t you need — leave the rest out. - Choose which variable to differentiate with respect to, and the order (1st, 2nd, or 3rd derivative w.r.t. that same variable).
- Optionally enter a numeric point — only the coordinates the resulting derivative actually depends on are used, so you can leave the rest blank.
- Read Partial Derivative for the symbolic result, Evaluated at Point for the numeric value at your point (if you gave one), and Step-by-Step Derivation for the full working.
Results update automatically as you type. For example, f = x^2*y + sin(x*y) differentiated with respect to x gives y * (2 * x + cos(x * y)), which evaluates to 3.16771 at x = 1, y = 2.
What is a partial derivative?
For a function of several variables, the partial derivative with respect to one variable measures how fast f changes as only that variable moves, with every other variable held fixed:
∂f/∂x = lim(h→0) [f(x+h, y, z, t) − f(x, y, z, t)] / hMechanically, this means every other variable is treated as a constant while you differentiate — so ∂/∂x (x²y) = 2xy, the same as differentiating x² alone and carrying the constant y through. A second-order partial (∂²f/∂x²) just repeats that process against the same variable again.
What functions can I enter?
Anything mathjs's expression evaluator understands: polynomials (x^2*y - 2*z), trigonometric functions (sin(x*z), cos(y)), roots and logarithms (sqrt(x), log(y)), and combinations of these, using x, y, z, and t as the four variables. Angles in trig functions are in radians by default — write pi for π.
Need the rate of change along an arbitrary direction instead of a single axis? See the Directional Derivative Calculator, which combines all the partials into a gradient. To evaluate arbitrary math expressions directly, see the Math Evaluator.
Privacy
This calculator runs entirely in your browser. Your function, variable choice, and point are never uploaded, logged, or stored — the computation happens locally on your device.