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Derivative Calculator — Step-by-Step (Chain / Product / Quotient Rule)

Derivative calculator with step-by-step — symbolic differentiation, chain/product/quotient rule shown, plot f(x) and f'(x).

  • Runs locally
  • Category Calculator
  • Best for Getting a realistic range before a purchase, plan, workout, or schedule decision.

Use x as the variable. Operators: + − * / ^. Functions: sin cos tan asin acos atan sinh cosh tanh ln log exp sqrt abs. Constants: pi e. No implicit multiplication — write 2*x, not 2x.

Result
cos(x^2) * 2 * x * exp(-x) + sin(x^2) * -exp(-x)
Steps (outermost rule first)
  1. product rule#1

    product rule on (sin(x^2))(exp(-x)): u'v + uv' = (cos(x^2) * 2 * x^1 * 1)(exp(-x)) + (sin(x^2))(exp(-x) * -1)

    d/dx sin(x^2) * exp(-x) → cos(x^2) * 2 * x^1 * 1 * exp(-x) + sin(x^2) * exp(-x) * -1
  2. chain rule#2

    chain rule on exp(-x): (exp(u))' = exp(u) · u'; here u = -x, u' = -1

    d/dx exp(-x) → exp(-x) * -1
  3. sign#3

    pulled the minus sign out: −(x)' = −(1)

    d/dx -x → -1
  4. variable#4

    d/dx of x is 1

    d/dx x → 1
  5. chain rule#5

    chain rule on sin(x^2): (sin(u))' = cos(u) · u'; here u = x^2, u' = 2 * x^1 * 1

    d/dx sin(x^2) → cos(x^2) * 2 * x^1 * 1
  6. power rule#6

    power rule: d/dx (x)^2 = 2 · (x)^1 · (x)' = 2 · (x)^1 · (1)

    d/dx x^2 → 2 * x^1 * 1
  7. variable#7

    d/dx of x is 1

    d/dx x → 1
f(x) and f'(x) on the same axes, x ∈ [−5, 5]
f(x)f'(x)
12 classic textbook exercises
Click any to load. Includes chain rule, product, quotient, and higher-order examples.
Expression, AST, derivative, plot — everything is computed in this browser tab.

What this tool does

A real symbolic derivative calculator, not a numerical approximator. Type any single-variable expression in x — sin(x^2) * exp(-x), ln(1 + x^2), 1 / (1 + x^2), x^3 + 2x - 7, whatever a textbook throws at you — and the page returns the derivative as a clean algebraic expression, plus the exact rule that was applied at every node of the tree (chain rule, product rule, quotient rule, power rule, plus the sixteen elementary-function derivatives). The steps are written the way a tutor explains them, not as a wall of LaTeX: "applied the chain rule, the outer function is sin so we get cos(x^2) and multiply by the derivative of x^2 which is 2x". Below the answer, an SVG plot draws f(x) and f'(x) on the same axes between x = −5 and 5 so you can see where the original is increasing, decreasing, and stationary. One-click second and third derivatives. Twelve classic textbook exercises (chain rule, products, quotients, implicit-style polynomials, higher-order) are bundled as presets so you can compare your handwork against a worked example. The expression parser, the symbolic differentiator, the simplifier, and the SVG renderer are all written from scratch in the browser tab — no math library, no server round trip, your input never leaves the page.

Tool details

Input
Text + Numbers
The page exposes text boxes, numeric controls, file pickers, or structured inputs depending on the tool.
Output
Live result + Copy + Preview
The result area focuses on usable output, with copy, download, or preview actions when supported.
Privacy
Browser-side processing
The main tool logic does not call an external API, so inputs normally stay in the current tab.
Save / share
Local preference storage
Preferences, history, or drafts are saved in this browser without an account.
Performance budget
Initial JS <= 32 KB
No WASM budget is declared, keeping the tool quick to open on mobile.
Best fit
Calculator · Student
Category and role tags drive related tools, internal links, and quick fit checks.

How to use

  1. 1. Input

    Paste or drop your content into the tool panel.

  2. 2. Process

    Click the button. All processing is local in your browser.

  3. 3. Copy / Download

    Copy the result or download to disk in one click.

How Derivative Calculator (Step-by-Step) fits into your work

Use it for fast estimates, comparisons, and planning numbers before you make the final call.

Calculation jobs

  • Getting a realistic range before a purchase, plan, workout, or schedule decision.
  • Comparing scenarios by changing one input at a time.
  • Turning rough assumptions into a number you can discuss.

Calculation checks

  • Double-check units, dates, rates, and rounding assumptions.
  • Treat health, finance, tax, and legal outputs as planning aids, not professional advice.
  • Save the inputs that produced an important result so you can reproduce it later.

Good next steps

These links move the current task into a more complete workflow.

  1. 1 Scientific Calculator Scientific calculator — sin / cos / log / sqrt / power, with full keyboard input + history, deg/rad mode. Open
  2. 2 Matrix Calculator Matrix calculator — add/sub/multiply/inverse/determinant/transpose/rank/RREF/eigenvalues for n×m matrices, step-by-step. Open
  3. 3 Quadratic Equation Solver Quadratic equation solver — ax² + bx + c = 0, discriminant analysis, real + complex roots, vertex/intercepts, parabola visualization. Open

Real-world use cases

  • Check a chain-rule homework answer before turning it in

    You wrote d/dx sin(x^2) = 2x cos(x^2) but the textbook key says cos(x^2) · 2x. Plug sin(x^2) into the calculator. It returns 2*x*cos(x^2) and the step panel reads "applied the chain rule with outer = sin, inner = x^2: outer'(inner) is cos(x^2), inner' is 2x, so the product is 2*x*cos(x^2)". Two minutes to confirm the order of multiplication does not matter and the cos(x^2) factor really is supposed to be inside, not outside, the parentheses.

  • See why product rule is not the same as multiplying derivatives

    A common slip on quizzes is to write (uv)' = u'v'. Type x^2 * sin(x). The step panel shows the product rule unfolding to 2x * sin(x) + x^2 * cos(x), and the simplified answer keeps both terms. The numerical plot then proves it: the curve of f' on the screen is not the curve of u'v', and seeing the two graphs is what makes the mistake stick in memory.

  • Read off where a function is increasing or decreasing

    For a max/min problem, paste your f(x) and look at the f' curve in the joint plot. Wherever f'(x) crosses zero is a horizontal tangent — a candidate for a local extremum. The calculator does not classify the critical point for you, but the visual is enough to spot the answer in seconds for most textbook functions.

  • Hand-verify a quotient-rule problem step by step

    The quotient rule (u/v)' = (u'v - uv') / v² is where signs and squared denominators eat marks. Type (x^2 + 1) / (x - 3). The step panel writes out u' = 2x, v' = 1, then assembles (2x*(x-3) - (x^2+1)*1) / (x-3)^2. You can copy the literal intermediate form to your worksheet and tick each substep — the calculator does not skip arithmetic to "look clean".

  • Get a second derivative without setting up the work twice

    You need f'' for a concavity question. Enter f, hit the "Second derivative" tab. The calculator runs the differentiator on its own result and gives you f''(x), with all the steps shown again from f' down to f''. Saves the manual rewrite and removes the most common transcription slip (forgetting a minus sign when copying f' onto the next line).

Common pitfalls

  • Writing implicit multiplication. "2x" and "sin x" do not parse — use 2*x and sin(x). The error message on the calculator points at the exact character where the parser stopped, so you don't have to hunt.

  • Confusing log and ln. On this calculator, log(x) means natural log (same as ln(x)). If you want base-10, write ln(x) / ln(10) explicitly. Textbooks disagree on the base for "log", so we picked one and document it; don't assume base-10.

  • Reading the simplified answer as the only valid form. The simplifier is intentionally conservative — it will not apply sin²+cos² = 1 or factor cleverly, because aggressive rewrites silently destroy the link between the answer and the rule that produced it. If your textbook simplifies further with an identity, both forms are right.

Privacy

The expression you type, the AST it parses into, the derivative that comes out, and every numeric sample on the plot are all computed inside your browser tab. Nothing is uploaded, logged, analyzed, or stored — not even in localStorage. Open the page, then disconnect the network: the calculator still works exactly the same on a phone in airplane mode or a classroom laptop with no internet. Refresh the tab and the page is back to a clean state with no record of what you computed.

FAQ

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Made by Toolora · 100% client-side · Updated 2026-06-13