Derivative Calculator
Symbolically differentiate any function of x — polynomials, trig, exponential and logarithmic expressions — and evaluate the derivative (the slope) at any point.
Function
Derivative f'(x)
3 · x^2 + 2 · 2 · x
d/dx [x^3 + 2x^2 - 5]
f(2)
11
f'(2) — slope
20
How It Works
A derivative measures how fast a function’s output changes as its input changes — geometrically, the slope of the tangent line at a point. This calculator parses your function into an expression tree and applies standard differentiation rules recursively: the sum rule (differentiate each term separately), the product rule ((uv)' = u'v + uv'), the quotient rule ((u/v)' = (u'v − uv')/v²), the generalized power rule ((uⁿ)' = n·uⁿ⁻¹·u') and the chain rule for nested functions like sin(x²). Built-in functions sin, cos, tan, exp (eˣ), ln (natural log), and sqrt each have known derivatives (e.g., d/dx[sin(x)] = cos(x)) that combine via the chain rule when their argument is itself an expression. The tool then simplifies the result and can evaluate both the original function and its derivative at any x-value you specify — useful for finding the slope of a curve at a specific point. Note: this calculator supports constant exponents (x³, x^0.5); variable exponents like xˣ require logarithmic differentiation and are out of scope.
Formula
Sum rule: (u+v)' = u'+v'. Product rule: (uv)' = u'v+uv'. Quotient rule: (u/v)' = (u'v−uv')/v². Power rule: (uⁿ)' = n·uⁿ⁻¹·u'. Chain rule applied throughout.
Frequently Asked Questions
What is a derivative in simple terms?
A derivative tells you the rate of change of a function — how much the output changes for a tiny change in input. Graphically, it’s the slope of the tangent line touching the curve at a specific point.
How do I find the derivative of x³ + 2x² − 5?
Differentiate term by term: d/dx[x³] = 3x², d/dx[2x²] = 4x, d/dx[−5] = 0. Adding them: f'(x) = 3x² + 4x. Enter "x^3 + 2x^2 - 5" into the calculator to verify.
What is the chain rule and when is it used?
The chain rule differentiates a function of a function: d/dx[f(g(x))] = f'(g(x)) · g'(x). It’s needed whenever a function’s argument is itself an expression, e.g., d/dx[sin(x²)] = cos(x²) · 2x.
Can this calculator handle x raised to a variable power (like xˣ)?
Not currently — that requires logarithmic differentiation (taking ln of both sides first). This tool supports constant exponents like x³ or x^0.5, which covers the vast majority of polynomial and algebraic functions.
What does it mean to evaluate the derivative at a point?
It gives the exact slope of the curve at that x-value. For f(x) = x², f'(x) = 2x, so at x = 3 the slope is 2×3 = 6 — the curve is rising steeply there. This is useful for finding tangent lines, rates of change, and optimization (where the slope is zero).