Instantaneous rate of change
The derivative measures how fast a function is changing at a single point — the slope of the tangent line, the velocity at an instant, the marginal cost at unit n.
f′(x) = lim Δx→0 [f(x+Δx) − f(x)] / Δx v1.0 · Symbolic, not numerical
Differentiate
anything,
step by step.
A free online derivative calculator that shows its work.
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Visualize
See f(x) and f′(x) side by side.
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Try
f(x)
x^3 - 2x + 1
f′(x)
-2 + 3·x^2
Concepts
What derivatives actually mean.
Tangent line
At every differentiable point, the derivative is the slope of the line that just touches the curve. Two curves meet at a tangent when their derivatives match.
y = f(x₀) + f′(x₀)·(x − x₀)Composition & chain rule
When functions are composed — sin(x²), e^(3x), ln(cos x) — the chain rule multiplies the outer derivative by the inner one, so the rate of change of the whole is the product of the parts.
d/dx f(g(x)) = f′(g(x)) · g′(x)Higher-order derivatives
Differentiating again gives the second derivative — the rate at which the rate is changing. Acceleration is the second derivative of position. Concavity is its sign.
f″(x) = d/dx [f′(x)] Why this calculator
Built for clarity, not just an answer.
Every step is recorded against the calculus rule that produced it.
∂
Symbolic engine
Parser, AST differentiator, simplifier, point evaluator.
∑
Step-by-step
Each rule applied is logged with the resulting expression.
ƒ
Evaluate anywhere
Type a value to see the derivative's numerical result instantly.
How it stacks up
A focused derivative calculator.
One thing, done well, with respect for your attention.
- →No login. The calculator works on the first visit, on any device.
- →No ads. Fast, single-purpose, with the result above the fold.
- →Symbolic, not numerical. Exact derivatives, simplified and printable.
- →Transparent steps. Every rule applied is recorded, not guessed.
derivative — bash
$ differentiate --f "x^3 - 2x + 1" --var x
→ step 1 power rule x^3 → 3·x^2
→ step 2 power rule −2x → −2
→ step 3 constant rule 1 → 0
→ step 4 sum rule a + b + c → 3·x^2 + (−2) + 0
f′(x) = 3·x^2 − 2
$ evaluate --at 4
f′(4) = 46 Deep Dive
The Ultimate Guide to Symbolic Differentiation
Calculus is the mathematical study of continuous change, and at the heart of calculus lies the derivative. Whether you are a student tackling homework or an engineer modeling physical systems, finding the rate of change is a fundamental task. Our derivative calculator online is designed to act not just as a tool that yields answers, but as a comprehensive educational companion. By using this interactive derivative calculator with steps, you can visualize, learn, and master the exact rules of differentiation in real time.
What is a Derivative and How Do You Find It?
Geometrically, the derivative of a function at a certain point is the slope of the tangent line to the graph of the function at that point. To find the derivative calculator users simply input an algebraic expression into the calculator. The internal engine parses the input, converts it into an Abstract Syntax Tree (AST), and differentiates it symbolically.
Historically, the derivative is defined using limits. Our definition of derivative calculator reference guide highlights the classic limit difference quotient:
f′(x) = lim h → 0 [ f(x + h) − f(x) ] / h
While evaluating this limit manually can involve tedious algebraic expansion (especially for complex trigonometric, fractional, or exponential terms), a symbolic first derivative calculator automates this by applying standard rules directly. These rules include the Power Rule, Product Rule, Quotient Rule, and the Chain Rule for composite functions.
Higher-Order Calculus: The Second and Double Derivative Calculator
Differentiating a function once gives you its first derivative, representing the instantaneous rate of change (like velocity). Differentiating the result a second time gives you the second derivative, representing the rate of change of the rate of change (like acceleration, or the concavity of a curve).
This tool functions as a fully automated second derivative calculator (also commonly referred to as a 2nd derivative calculator or double derivative calculator). To compute a higher-order derivative, the engine performs sequential passes:
- It parses and differentiates the original expression to get the first derivative.
- It simplifies the intermediate result.
- It differentiates the simplified intermediate expression to yield the second derivative.
This sequence is repeated for any order you select (1st, 2nd, or 3rd order in the UI, and even higher for custom equations).
Multivariable Calculus: Partial and Directional Derivatives
In real-world applications, functions often depend on more than one variable. For example, the temperature of a metal plate might depend on both its horizontal and vertical coordinates, represented as f(x, y).
- → Partial Derivative Calculator: When you differentiate a multivariable function with respect to one variable while holding all other variables constant, you are finding a partial derivative. For example, ∂f/∂x represents the rate of temperature change along the x-axis. In this calculator, you can select variables like
x,yto compute mixed partial derivatives step-by-step. - → Directional Derivative Calculator: The directional derivative measures the rate of change of a function in the direction of an arbitrary unit vector. While the calculator doesn't directly take a vector input, it computes the gradient (the vector of all partial derivatives: ∇f = <fx, fy>). You can find the directional derivative by taking the dot product of this gradient with your unit vector: Duf = ∇f · u.
Implicit Differentiation: Solving Complex Relations
Not all mathematical relations are defined explicitly in the form y = f(x). For equations like x² + y² = 1 (a circle) or y² − x² = 4 (a hyperbola), y is defined implicitly in terms of x.
To find the rate of change dy/dx for these relations, you use an implicit derivative calculator. The tool accomplishes this by rewriting the equation as F(x, y) = 0, finding the partial derivatives with respect to x and y, and applying the implicit function theorem:
dy/dx = − ( ∂F/∂x ) / ( ∂F/∂y )
This method allows the calculator to find derivatives of complex curves without needing to solve for y first, showing every step of the partial derivatives and combining them automatically.
Try it on a function you actually care about.
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