Derivative Calculator — Basic, Implicit, Partial, nth-Order & Antiderivatives
Type any function and get an instant, fully worked derivative — or antiderivative — with every rule applied and explained. Built for students, engineers, and anyone who needs to see the working, not just the answer.
Step-by-Step Solution
Whether you're checking a homework answer, working through an engineering problem set, or trying to remember how the chain rule actually applies to sin(x²), the AceCalculator derivative calculator gives you an answer you can trust — and, more importantly, shows you exactly how it got there.
Every result is built from the same rules you'd apply by hand: the power rule, product rule, quotient rule, and chain rule. AceCalculator applies them in order, labels each one, and lets you see the calculation unfold instead of just handing you a final expression.
This tool covers five distinct modes in one place — ordinary derivatives, implicit differentiation, partial derivatives, higher-order (nth) derivatives, and antiderivatives — so you don't need five different calculators bookmarked. Everything runs directly in your browser. No downloads, no accounts, no data sent anywhere.
What You Can Do With This Calculator
Basic Derivatives — Polynomials, Trig, Exponentials & Logs
The basic derivative mode handles single-variable functions of any complexity: polynomials, roots, trig functions, exponentials, and natural or common logarithms, combined freely.
- Power, sum, and difference rules for polynomials
- Product and quotient rules for combined functions
- Chain rule for nested and composite functions
- Trig, inverse trig, exponential, and logarithmic derivatives
Implicit Differentiation
When y can't be isolated on one side of the equation, the implicit differentiation calculator differentiates both sides with respect to x, treats y as a function of x, and algebraically solves for dy/dx.
- Works on equations like x² + y² = 25 or xy = 1
- Applies the chain rule automatically to every y-term
- Isolates and simplifies dy/dx as the final step
Partial Derivatives
For multivariable functions, the partial derivative calculator differentiates with respect to one variable while holding the others constant — with support for mixed second-order partials.
- ∂f/∂x, ∂f/∂y, and ∂f/∂z for multivariable functions
- Mixed partials such as ∂²f/∂x∂y
- Clear notation showing which variables are held constant
nth-Order Derivatives
Need the second, third, or sixth derivative? The nth-order derivative calculator differentiates repeatedly and shows the result at every order along the way, not just the last one.
- Choose any order from 1st to 6th derivative
- See every intermediate derivative, fully simplified
- Useful for Taylor series, concavity, and motion problems
Antiderivatives (Indefinite Integrals)
The antiderivative calculator reverses the process, applying the power rule, standard integral forms, and reverse chain rule for linear substitutions — always with the constant of integration.
- Power, exponential, logarithmic, and trig integral forms
- Reverse chain rule for functions like sin(3x + 1)
- Every result includes "+ C"
Full Step-by-Step Breakdown
Every mode returns a numbered list of steps, each one tagged with the specific rule used — Power Rule, Product Rule, Quotient Rule, Chain Rule, or a standard integral form — plus the formula being applied.
- Rule-by-rule breakdown, not just a final answer
- Formulas shown alongside each applied step
- Designed to match how the topic is taught in class
Other Derivative Problems You Can Work Through Here
A lot of derivative questions don't fall neatly under "basic" or "partial" — here's how the modes above map onto some of the specific problems students search for most.
Chain Rule Problems
Nested functions like sin(x²) or (3x + 1)⁵ are where the chain rule trips people up, because it's easy to forget the inner derivative. Type the composite function straight into Basic mode and the outer and inner derivatives are shown as separate, labelled steps before they're multiplied together.
- Works for functions nested two or three layers deep
- Chain rule steps are kept separate from power or product rule steps
- Combines automatically with product and quotient rules when needed
Logarithmic Differentiation
Functions where the variable appears in both the base and the exponent — like xˣ or (sin x)ˣ — can't be handled by the power rule alone. The standard fix is to take ln of both sides, differentiate implicitly, then multiply back through by the original function. Once you've taken that ln step by hand, enter the resulting expression into Basic mode to get the rest of the working.
- Covers the "ln both sides" technique used for variable-exponent functions
- Also useful for differentiating long products or quotients of several terms
Second Derivatives & Higher-Order Derivatives
Set nth-Order mode to n = 2 for the second derivative, or go higher for concavity, inflection point, or Taylor series work. Every intermediate order is shown, so you can see the first derivative on the way to the second, not just the final result.
- 2nd, 3rd, or up to 6th derivative in one calculation
- Every intermediate order shown and simplified
Derivative at a Specific Point
Differentiate your function symbolically in Basic mode first, then substitute your x-value into the resulting expression. If the arithmetic gets messy, our Scientific Calculator handles the numeric evaluation once you've got the derivative in hand.
- Get the full symbolic derivative before plugging in a value
- Useful for slope-of-tangent-line and instantaneous rate of change problems
Multivariable & Directional Derivatives
Directional derivatives are built from partial derivatives — you take ∂f/∂x and ∂f/∂y in Partial mode, then combine them with your direction vector. This calculator gives you the exact partial-derivative components (the gradient) you need for that final step.
- ∂f/∂x, ∂f/∂y, ∂f/∂z with full working
- Mixed second-order partials for gradient and Hessian work
Derivatives of Inverse Functions
The keypad includes sin⁻¹, cos⁻¹, and tan⁻¹ directly, so inverse trig derivatives are handled the same way as any other Basic mode function — the standard inverse-function derivative formulas are applied and shown as a labelled step.
- arcsin, arccos, and arctan derivatives built in
- Steps show the formula used, not just the result
Parametric Equation Derivatives
For x(t) and y(t) pairs, dy/dx = (dy/dt) ÷ (dx/dt). Differentiate x(t) and y(t) separately in Basic mode with respect to t, then divide the two results by hand to get dy/dx.
- Differentiate each parametric component with full steps
- Works for any t-parameterised curve
The Rules Behind the Calculator
Every step this calculator produces comes from one of the standard differentiation or integration rules below. Understanding these is the fastest way to check the working by hand.
| Rule | Formula | Used For |
|---|---|---|
| Power Rule | d/dx[xⁿ] = n·x⁽ⁿ⁻¹⁾ | Polynomial terms |
| Sum / Difference Rule | d/dx[f ± g] = f′ ± g′ | Multi-term expressions |
| Product Rule | d/dx[f·g] = f′g + fg′ | Two functions multiplied |
| Quotient Rule | d/dx[f/g] = (f′g − fg′)/g² | One function divided by another |
| Chain Rule | d/dx[f(g(x))] = f′(g(x))·g′(x) | Nested / composite functions |
| Power Rule for Integrals | ∫xⁿ dx = x⁽ⁿ⁺¹⁾/(n+1) + C | Antiderivatives of polynomial terms |
Numerical Differentiation Formulas (Reference)
This calculator finds derivatives symbolically, using the exact rules above rather than approximation. If you're working from tabulated data instead of a formula, these are the standard finite-difference formulas used to estimate a derivative numerically.
| Method | Formula | Notes |
|---|---|---|
| Forward Difference | f′(x) ≈ [f(x+h) − f(x)] / h | Simplest estimate, uses the next point |
| Backward Difference | f′(x) ≈ [f(x) − f(x−h)] / h | Uses the previous point instead |
| Central Difference | f′(x) ≈ [f(x+h) − f(x−h)] / 2h | More accurate — errors cancel on both sides |
| Central Difference (2nd derivative) | f″(x) ≈ [f(x+h) − 2f(x) + f(x−h)] / h² | Standard three-point second-derivative estimate |
Why AceCalculator Is Different And Better
Most derivative calculators online give you an answer and nothing else. AceCalculator focuses on showing the working, matching classroom notation, and staying fast on any device.
- ✓ Full rule-by-rule step-by-step solutions, every mode
- ✓ Five calculators in one: basic, implicit, partial, nth-order, antiderivative
- ✓ Clean, classroom-style notation
- ✓ Mobile-friendly and responsive on any device
- ✓ No downloads, no accounts, no signup required
- ✓ Runs entirely in your browser — nothing is stored or sent to a server
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Frequently Asked Questions
Yes. Every mode — basic, implicit, partial, nth-order, and antiderivative — is free to use with no account, signup, or download required.
Yes. Each result comes with a numbered list of steps, and each step is labelled with the specific rule applied (Power Rule, Product Rule, Quotient Rule, or Chain Rule) along with the formula used.
Basic differentiation works on a function already written as y = f(x). Implicit differentiation is used when x and y are mixed together in an equation (like x² + y² = 25) and y can't easily be isolated — you differentiate both sides and solve for dy/dx.
Yes. Switch to Partial mode, enter a multivariable function, and choose which variable to differentiate with respect to. Mixed second-order partials (like ∂²f/∂x∂y) are also supported.
Yes. Antiderivative mode covers the standard integral forms — power rule, exponential, logarithmic, and trig integrals — plus reverse chain rule for functions with a linear inner term, always including the constant of integration, C.
The calculator applies the same differentiation and integration rules taught in standard calculus courses. As with any automated tool, always double-check results for high-stakes coursework or professional work.
Those tools are solid options too, and this calculator is built with the same goal — a fast, accurate derivative with visible working. The difference here is that it's free with no account or paywall on the step-by-step solutions, and it runs entirely client-side, so nothing you type is sent to a server.
Yes. Graphing calculators like the TI-84 or Casio fx series can compute a numeric derivative at a point, but they generally won't show you the symbolic, rule-by-rule working. This tool is a good way to cross-check the algebra behind whatever your calculator gives you.
Related rates problems usually require you to differentiate an equation implicitly with respect to time — Implicit mode (using t as the variable) handles that piece. Continuity of piecewise functions is a separate check (matching left- and right-hand limits at the boundary points) that this calculator doesn't perform directly, though a function needs to be continuous at a point before its derivative there means anything.
Important Disclaimer
AceCalculator's derivative calculator is provided for informational and educational purposes only. While every step follows standard differentiation and integration rules, results are not a substitute for verification by a qualified instructor or professional where accuracy is critical (exams, published research, or engineering work). See our full Disclaimer and Terms and Conditions.