Derivative Calculator
Enter a function of x.
Try an example
f(x) = x³ + 2x² − 5x
f'(x) = 3x² + 4x − 5
Rules Used
Power Rule
d/dx(xⁿ) = n·xⁿ⁻¹
Sum Rule
d/dx(f + g) = f' + g'
Constant Rule
d/dx(c·f) = c·f', d/dx(c) = 0
Step-by-Step Solution
Step 1 — Differentiate each term
d/dx(x³) + d/dx(2x²) + d/dx(−5x)
Step 2 — Apply the rules to each term
d/dx(x³) = 3x²
d/dx(2x²) = 4x
d/dx(5x) = 5
Step 3 — Combine the results
f'(x) = 3x² + 4x − 5
Evaluate derivative at a point
Enter an x-value to evaluate f'(x).
Function and Derivative
Critical Points
Solve f'(x) = 0
This derivative calculator finds the derivative of a function - the rate at which the function's output changes relative to its input - using standard differentiation rules like the power rule, product rule, quotient rule, and chain rule.
Enter a function, and the calculator returns its derivative along with the rule(s) applied.
Worked Calculation Examples
| Scenario | Result | Calculation Step |
|---|---|---|
| f(x) = 3x² + 5x, find f'(2) | f'(x) = 6x + 5, so f'(2) = 17 | Power rule on each term: d/dx[3x²] = 6x, d/dx[5x] = 5. f'(x) = 6x + 5. At x = 2: 6(2) + 5 = 17. |
| f(x) = sin(x²), find f'(1) | f'(x) = 2x·cos(x²), so f'(1) ≈ 1.08 | Chain rule: derivative of the outer function (cos) times derivative of the inner function (2x) gives f'(x) = 2x·cos(x²). At x = 1: 2(1)·cos(1) ≈ 2 × 0.5403 ≈ 1.08. |
What a Derivative Represents
A derivative measures instantaneous rate of change - graphically, it's the slope of the tangent line to a function's curve at any given point. In applied terms, if a function describes position over time, its derivative describes velocity; the derivative of velocity in turn describes acceleration.
Common Differentiation Rules
The power rule handles terms like x^n (derivative: n*x^(n-1)). The product rule handles functions multiplied together, the quotient rule handles functions divided by each other, and the chain rule handles composite functions (a function inside another function) - most derivative problems combine two or more of these rules.
How to Use the Derivative Calculator
- Enter a function of x.
- Review the formatted function.
- Calculate the derivative.
- Open the step-by-step solution to see which rules were applied.
- Enter an x-value to evaluate the derivative at that point.
- Use the graph to visualize the function, derivative, and tangent line.
Frequently Asked Questions
What is a derivative in calculus?
A derivative measures the instantaneous rate of change of a function - graphically, the slope of the line tangent to the function's curve at a specific point.
What is the power rule for derivatives?
For a term x^n, the derivative is n*x^(n-1) - multiply by the original exponent, then reduce the exponent by one.
When do I need the chain rule?
Use the chain rule whenever you're differentiating a composite function (one function nested inside another, like sin(x^2)) - multiply the derivative of the outer function by the derivative of the inner function.
How do I use this derivative calculator?
Enter the known values, review the units or settings, and the calculator updates the result instantly. The formula and example on this page show how the answer is produced.
What does the Derivative Calculator calculate?
Solve Derivative problems online with fast math inputs and results. It is designed for fast browser-based calculations without sign-up, downloads, or manual spreadsheet setup.
What formula does this derivative use?
The formula is: d/dx [xⁿ] = n·xⁿ⁻¹ (power rule, plus product, quotient, and chain rules for composite functions). Isaac Newton and Gottfried Leibniz independently developed the foundations of differential calculus in the late 17th century, and famously disputed credit for the discovery for the rest of their lives - a controversy historians now generally resolve by crediting both with independent invention. Leibniz's notation (dy/dx), rather than Newton's, became the standard used in mathematics today because it more clearly shows the relationship between the changing quantities involved.
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Last updated: September 27, 2026.