Derivative Calculator

Compute symbolic derivatives of any function of x (polynomials, trig, exponentials, logs). Visualize original function and its derivative on an interactive graph.

Use standard math notation: ^ for exponent, * for multiplication, sin, cos, tan, ln (natural log), exp(x), sqrt(x).
Quick examples:
x²
sin(x)
cos(x)
eˣ
ln(x)
tan(x)
x³ - 3x
x·sin(x)
Privacy first: All calculations happen locally using nerdamer symbolic engine. No data leaves your browser.
Computation note: This tool is designed for explicit single-variable functions. For functions involving absolute values (abs(x)), piecewise definitions, or requiring implicit or partial differentiation, specialized tools may be needed. We continuously update to expand supported function types.

What Is a Derivative? Calculus at Your Fingertips

The derivative of a function measures the instantaneous rate of change — the slope of the tangent line at any point. In Leibniz notation, dy/dx. Our derivative calculator uses symbolic computation (powered by nerdamer) to deliver exact derivatives for any elementary function. From physics (velocity, acceleration) to economics (marginal cost), derivatives are everywhere.

Intuitive understanding: Imagine driving a car. The speedometer shows your instantaneous speed—the derivative of your position with respect to time. In economics, the derivative of a cost function (marginal cost) tells you the additional cost of producing one more unit. The derivative is the universal mathematical language for this instantaneous rate of change.

\[ f'(x) = \lim_{h \to 0} \frac{f(x+h)-f(x)}{h} \]

The formal definition of derivative — our calculator automates the algebra behind it.

How to Use the Derivative Calculator: A Step‑by‑Step Example

Let's analyze the function f(x) = x³ − 3x to see how the calculator helps find critical points and understand function behavior:

  1. Input: Enter "x^3 - 3*x" in the function box.
  2. First derivative: Click "Differentiate & Plot". The calculator shows f'(x) = 3*x² − 3, which gives the slope at any point x.
  3. Find critical points: Set the derivative to zero: 3*x² − 3 = 0 → x = 1 or x = −1. These are points where the slope is zero (potential local maxima, minima, or inflection points).
  4. Second derivative: Select "2nd derivative" from the dropdown. The calculator returns f''(x) = 6*x, which indicates concavity.
  5. Graphical verification: Observe the graph. The original function (black curve) has a local maximum at x = −1 and a local minimum at x = 1. The derivative (orange curve) crosses zero at these points. The second derivative (not drawn but computable) is negative for x < 0 (concave down) and positive for x > 0 (concave up), confirming the nature of the critical points.

This workflow demonstrates how the calculator can verify manual work, explore function behavior, and connect algebraic results with visual intuition.

Why Higher-Order Derivatives Matter

The second derivative (f''(x)) reveals concavity and inflection points. The third derivative appears in Taylor series expansions and in physics as jerk (rate of change of acceleration). Our calculator supports up to third order, helping you analyze function behavior comprehensively.

Real‑World Application: Optimization & Marginal Analysis

A manufacturer models daily profit as P(x) = -2x³ + 150x² - 1200x + 5000 (x = units). The derivative P'(x) gives marginal profit — the additional profit from producing one more unit. Setting P'(x)=0 finds critical points (max profit). Our calculator instantly computes P'(x) = -6x² + 300x - 1200, and P''(x) = -12x + 300 to confirm maxima. Engineers and economists rely on such differentiation for decision making.

Derivative Rules & Techniques — Quick Reference

Rule Expression Derivative
Power Rule xⁿ n·xⁿ⁻¹
Product Rule u·v u'v + uv'
Quotient Rule u/v (u'v - uv')/v²
Chain Rule f(g(x)) f'(g(x))·g'(x)
Trigonometric sin(x) cos(x)
Exponential eˣ eˣ
Logarithmic ln(x) 1/x

Frequently Asked Questions

All elementary functions: polynomials, trigonometric (sin, cos, tan, sec, csc, cot), exponentials (exp, e^x), logarithms (ln, log), square roots (sqrt), and compositions. It also handles implicit multiplication.

Symbolic differentiation is exact — the tool applies derivative rules algebraically, so there is no numerical approximation error. The displayed expression is simplified by nerdamer’s CAS engine.

This version focuses on explicit functions of x. For partial derivatives or implicit differentiation, consider specialized tools, but our calculator handles all standard single‑variable cases.

The graph displays over x ∈ [−5,5]. Functions undefined in that interval (e.g., ln(x) for negative x) will show gaps. The derivative might also have vertical asymptotes. The tool gracefully handles domain issues.

Absolutely. Students can verify their manual differentiation, check critical points, and see graphical connections between f and f'. It's a great learning companion.

Academic foundation: Built upon rigorous symbolic differentiation algorithms. The tool uses nerdamer – an open‑source computer algebra system (CAS) for JavaScript that strictly implements differentiation rules as defined in standard calculus textbooks (Stewart, Thomas). Its algorithms undergo extensive mathematical testing and are used in educational technology projects worldwide. Reviewed by getzenquery Tech team. Updated April 2026.

References: Wolfram MathWorld – Derivative; nerdamer documentation; OpenStax Calculus Volume 1.