Function Graph Explorer

Plot functions, explore their properties, and see step-by-step derivative analysis. Updated in real-time.

Functions

f
g
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Properties

Enter a function and click Graph to see properties.

Quick Reference

Derivative Rules

Power Rule
ddx[xn]=nxn−1\frac{d}{dx}[x^n] = nx^{n-1}
Product Rule
ddx[f⋅g]=f′g+fg′\frac{d}{dx}[f \cdot g] = f'g + fg'
Chain Rule
ddx[f(g(x))]=f′(g(x))⋅g′(x)\frac{d}{dx}[f(g(x))] = f'(g(x)) \cdot g'(x)

Trig Derivatives

ddx[sin⁡x]=cos⁡x\frac{d}{dx}[\sin x] = \cos x
ddx[cos⁡x]=−sin⁡x\frac{d}{dx}[\cos x] = -\sin x
ddx[tan⁡x]=sec⁡2x\frac{d}{dx}[\tan x] = \sec^2 x
ddx[ex]=ex\frac{d}{dx}[e^x] = e^x
ddx[ln⁡x]=1x\frac{d}{dx}[\ln x] = \frac{1}{x}

Graph Features

Roots Where f(x)=0f(x) = 0. The function crosses the x-axis.
Extrema Where f′(x)=0f'(x) = 0. Local maxima or minima.
Inflection Where f′′(x)=0f''(x) = 0. Concavity changes.

Increasing: f′(x)>0f'(x) > 0  |  Decreasing: f′(x)<0f'(x) < 0

Interval Notation

Open interval (a,b)(a, b) Excludes endpoints.
Closed interval [a,b][a, b] Includes endpoints.
Unbounded (−∞,a)(-\infty, a) or (b,∞)(b, \infty)
Union A∪BA \cup B Combines intervals.
Example (−∞,−1)∪(2,∞)(-\infty, -1) \cup (2, \infty) means all xx less than −1-1 or greater than 22.

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Related Labs

Derivatives & Tangent Lines Lab
Visualize derivatives as slopes of tangent lines, animate secant lines approaching tangents as h approaches 0, explore derivative rules step by step, and compute slopes on implicit curves
Limits & Continuity Lab
Explore the formal epsilon-delta definition of limits, apply the squeeze theorem, and classify removable, jump, and infinite discontinuities by testing continuity conditions
Function Transformations Lab
Explore how a, b, h, and k transform parent functions. Shift, stretch, reflect, and find inverses for linear, quadratic, absolute value, square root, cubic, and reciprocal functions
Polynomial & Rational Functions Lab
Analyze polynomial roots, end behavior, and turning points. Explore rational functions with vertical/horizontal/oblique asymptotes and holes using interactive JSXGraph plots

Study Posters

Derivative Rules Visual Guide
Derivative Rules Visual Guide
Calculus
Tangent Lines and Linear Approximation
Tangent Lines and Linear Approximation
Calculus
Curve Sketching with Derivatives
Curve Sketching with Derivatives
Calculus
First Derivative Test
First Derivative Test
Calculus

Worksheets

Calculus: Derivative Rules: Power, Chain, Product, Quotient
9-12
Piecewise and Step Functions
9-12
Quadratic Functions and Parabolas
9-12
Exponential Functions and Growth
9-12

Cheat Sheets

Derivatives & Differentiation Rules
Derivatives & Differentiation Rules
CalculusGrade 11-12
Applications of Calculus
Applications of Calculus
CalculusGrade 11-12
Limits & Continuity
Limits & Continuity
CalculusGrade 11-12
Integration Techniques
Integration Techniques
CalculusGrade 11-12