liminfoDerivative Calculator
Free web tool: Derivative Calculator
Supported: x^n, sin(x), cos(x), tan(x), e^x, ln(x), constants
f'(x) =
6x + 2cos(x) - 1e^x
Steps
- d/dx[3x^2] = 2 × 3 × x^(2-1) = 6x (power rule)
- d/dx[2sin(x)] = 2cos(x)
- d/dx[-1e^x] = -1e^x
- d/dx[5] = 0 (constant rule)
Derivative Rules
About Derivative Calculator
The Derivative Calculator is a free, browser-based tool that symbolically differentiates mathematical expressions and shows every differentiation step in detail. It supports polynomial terms (x^n), trigonometric functions (sin, cos, tan), the natural exponential (e^x), the natural logarithm (ln(x)), and scalar constants — all combined in a single expression like "3x^2 + 2sin(x) - e^x + 5".
Calculus students, high school and university educators, engineers, and data scientists use this tool when they need to verify derivative calculations quickly without pen-and-paper work. The calculator applies the power rule, constant rule, and the standard derivatives of sin, cos, tan, e^x, and ln(x) term by term, making it easy to follow the logic behind each result.
All computation happens entirely inside your web browser using JavaScript. There is no backend server, no data upload, and no account required. The interface adapts to dark mode and is fully responsive on phones, tablets, and desktop screens — making it a reliable study companion wherever you are.
Key Features
- Symbolic differentiation of polynomials: applies d/dx[x^n] = nx^(n-1) (power rule)
- Trigonometric derivatives: sin(x) → cos(x), cos(x) → -sin(x), tan(x) → sec²(x)
- Exponential derivative: d/dx[e^x] = e^x with coefficient scaling support
- Logarithmic derivative: d/dx[ln(x)] = 1/x, with multi-coefficient expressions
- Constant rule: d/dx[c] = 0 for any numeric constant term
- Step-by-step breakdown listing the rule applied to each term individually
- Live preview — result updates instantly as you type the expression
- Supports mixed multi-term expressions with addition and subtraction
Frequently Asked Questions
What functions does the Derivative Calculator support?
The calculator supports polynomial terms (x^n for any real power n), the trigonometric functions sin(x), cos(x), and tan(x), the natural exponential e^x, the natural logarithm ln(x), and numeric constants. You can combine them freely in a single expression.
How does the calculator handle multi-term expressions?
The input parser splits the expression on "+" and "-" signs into individual terms, differentiates each one independently using the appropriate rule, drops zero-derivative constants, and then joins the remaining results back together into a final derivative expression.
What differentiation rules are applied?
The calculator uses the constant rule (d/dx[c] = 0), the power rule (d/dx[x^n] = nx^(n-1)), and the standard derivatives of sin, cos, tan, e^x, and ln(x). The constant multiple rule (d/dx[cf(x)] = cf'(x)) is applied automatically to scale coefficients.
Can the calculator differentiate a function like 3x^2 + 2sin(x) - e^x + 5?
Yes. The calculator parses each term: 3x^2 becomes 6x (power rule), 2sin(x) becomes 2cos(x), -e^x stays -e^x, and the constant 5 becomes 0. The combined result is "6x + 2cos(x) - e^x".
Does it support chain rule or product rule?
Not in the current version. The calculator differentiates each term independently, which covers single-variable polynomial, trig, exponential, and logarithmic terms. Composite functions requiring the chain rule or products requiring the product rule would need a more advanced symbolic math engine.
Why does the result update without pressing a button?
The tool uses React's useMemo hook to recompute the derivative every time the input changes. The parsing and differentiation run entirely in memory with no network round-trips, so results appear in milliseconds.
How should I enter a term like negative 2x cubed?
Type "-2x^3". The parser treats the leading minus sign as a negative coefficient and extracts the power from the "^" notation. The derivative will be computed as d/dx[-2x^3] = -6x^2.
Is this tool accurate for exam verification?
Yes, for the supported function types the calculator applies the same rules taught in standard calculus courses. It is well-suited for checking homework and exam answers, but always verify composite-function results manually since chain rule is not yet supported.