Derivative Calculator
Symbolically differentiate any mathematical function with step-by-step explanations. Supports polynomials, trigonometric, exponential, logarithmic, and inverse trig functions using the power rule, product rule, quotient rule, and chain rule. Export LaTeX with one click.
Enter a mathematical expression and instantly compute its derivative with step-by-step explanation.
Use ^ for powers, * for multiplication. Supported: sin, cos, tan, sec, csc, cot, ln, exp, sqrt, asin, acos, atan.
Differentiate with respect to
d/dx [x^3 + 2*x^2 - 5*x + 7]
Derivative simplified using standard calculus rules
x^3 + 2*x^2 - 5*x + 7
3 * x^2 + 4x - 5
\frac{d}{dx}\left[{x}^{3} + 2 \cdot {x}^{2} - 5x + 7\right] = 3 \cdot {x}^{2} + 4x - 5Why Use Our Derivative Calculator?
Symbolic Differentiation Engine
Parses your expression into an Abstract Syntax Tree (AST) and applies calculus rules symbolically — giving exact algebraic derivatives, not numerical approximations.
All Calculus Rules Supported
Handles the power rule, product rule, quotient rule, chain rule, sum/difference rule, and constant multiple rule — automatically selecting the correct rule for each sub-expression.
Step-by-Step Explanations
Toggle the step-by-step breakdown to see exactly which rule was applied at each stage, along with inner and outer derivative sub-steps for complex functions.
100% Client-Side Privacy
Every parse, differentiation, and simplification runs entirely in your browser. No expression is ever sent to a server — your work stays completely private.
LaTeX Copy-Ready Output
Export your result as a ready-to-paste LaTeX snippet for use in Overleaf, Notion, Jupyter notebooks, or academic papers with a single click.
Automatic Simplification
Results are simplified using iterative constant folding and algebraic identities (x·1 = x, x+0 = x, 0·x = 0) to produce the most readable form.
Who Uses This Derivative Calculator?
Students & Homework Verification
Verify calculus homework, check worked solutions, and learn which rule applies to each expression before your exam.
Software & Algorithm Development
Engineers implementing gradient descent or optimization loops can verify analytical gradient expressions against their code.
Economic & Financial Modelling
Economists and analysts can compute marginal cost, revenue, and utility functions by differentiating rational polynomial models.
Tutoring & Classroom Teaching
Math tutors can generate example derivative problems with full rule breakdowns to illustrate product/quotient/chain rule concepts.
Physics & Engineering
Scientists can differentiate velocity to obtain acceleration, or compute the rate of change for any physical law expression.
Machine Learning Research
Researchers can derive gradients of activation functions or loss functions analytically and validate automatic differentiation outputs.
Understanding Symbolic Differentiation & Calculus Rules
What Is a Derivative?
A derivative measures the instantaneous rate of change of a function with respect to one of its variables. Formally, the derivative of f(x) with respect to x is defined as the limit: f′(x) = lim(h→0) [f(x+h) − f(x)] / h. Geometrically, it represents the slope of the tangent line to the curve y = f(x) at any given point. Derivatives are fundamental to calculus and have applications in physics, economics, machine learning, and virtually every field of science and engineering.
Power Rule & Sum/Difference Rule
The power rule states that d/dx[xⁿ] = n·xⁿ⁻¹ for any real constant n. It is the most frequently used differentiation rule and extends via the chain rule so that d/dx[u(x)ⁿ] = n·u(x)ⁿ⁻¹·u′(x). The sum rule states that the derivative of a sum of functions equals the sum of their individual derivatives: d/dx[f + g] = f′ + g′. The difference rule works analogously: d/dx[f − g] = f′ − g′. These rules allow you to differentiate polynomials term-by-term without needing more complex techniques.
Product Rule & Quotient Rule
When differentiating a product of two functions, use the product rule: d/dx[u·v] = u′·v + u·v′. This is essential whenever neither factor is constant. For a quotient u/v, the quotient rule gives: d/dx[u/v] = (u′·v − u·v′) / v². A helpful mnemonic is "low·d(high) minus high·d(low), all over low squared." When the denominator is a constant, the quotient rule reduces to the simpler constant multiple rule: d/dx[u/c] = u′/c.
Chain Rule for Composite Functions
The chain rule is used when differentiating a composition of two functions f(g(x)). It states: d/dx[f(g(x))] = f′(g(x)) · g′(x). In words: differentiate the outer function, leaving the inner function unchanged, then multiply by the derivative of the inner function. For example, d/dx[sin(x²)] = cos(x²) · 2x. This rule is the foundation for differentiating all composite expressions and implicitly underlies every trigonometric, logarithmic, and exponential derivative when the argument is not just x.
Trigonometric & Transcendental Derivatives
Standard transcendental derivatives include: d/dx[sin(x)] = cos(x), d/dx[cos(x)] = −sin(x), d/dx[tan(x)] = sec²(x), d/dx[eˣ] = eˣ, d/dx[ln(x)] = 1/x, and d/dx[√x] = 1/(2√x). Inverse trig functions: d/dx[arcsin(x)] = 1/√(1−x²), d/dx[arccos(x)] = −1/√(1−x²), d/dx[arctan(x)] = 1/(1+x²). Combined with the chain rule, these cover the full suite of elementary functions encountered in single-variable calculus. This calculator supports all of these natively.
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Derivative Calculator — Frequently Asked Questions
The calculator handles polynomials, rational functions, trigonometric functions (sin, cos, tan, sec, csc, cot), inverse trig (asin, acos, atan), natural logarithm (ln), exponential (exp, e^x), square root (sqrt), and arbitrary compositions and products of these. You can write 2x^3 + sin(x)*ln(x) or (x^2+1)/(x-1) and it will work correctly.
Use ^ for exponentiation (e.g. x^3), * for multiplication (e.g. 2*x or x*sin(x)), and standard parentheses for grouping. Function names are lowercase: sin(x), cos(x), tan(x), ln(x), exp(x), sqrt(x), asin(x), acos(x), atan(x). You can omit the multiplication sign between a number and a variable (e.g. 3x is valid) or between a number and a function (e.g. 2sin(x)).
This is full symbolic differentiation. The calculator parses your expression into an Abstract Syntax Tree (AST) and applies analytical calculus rules (power rule, product rule, chain rule, etc.) to produce an exact algebraic derivative — not a finite-difference numerical approximation. The result is a simplified closed-form expression.
Yes. The engine recursively applies the chain rule for arbitrarily nested compositions. For example, sin(x^2), ln(cos(x)), exp(sin(x^3)), or sqrt(x^2+1) are all handled correctly by differentiating the outer function and multiplying by the derivative of the inner function at each nesting level.
Yes. Use the "Variable" input field to set your differentiation variable to any single-letter identifier such as t, y, or r. All other letters in the expression will then be treated as constants. For example, differentiating a*t^2 + b*t with respect to t gives 2*a*t + b.
After computing the raw derivative AST, the tool runs an iterative simplification pass that folds constant sub-expressions (e.g. 3*2 → 6), removes additive/multiplicative identities (x+0 = x, x*1 = x, x*0 = 0), commutes constants to the left (x*2 → 2x), groups nested constants, and eliminates double negations. This produces the most readable canonical form.
The LaTeX output is formatted as a standard derivative notation: \frac{d}{dx}[...] = ... You can paste it directly into Overleaf, any Markdown editor that supports math (Notion, Obsidian, GitHub Markdown with MathJax), Jupyter notebooks, or academic paper templates. Click "Copy LaTeX" to copy it to your clipboard.
Yes, the Derivative Calculator is completely free to use with no account, signup, or usage limits. All parsing and differentiation happens entirely inside your browser using JavaScript. Your expressions are never transmitted to any server, ensuring your mathematical work stays completely private.