Derivative Calculator with Complete Function Analysis

Differentiate functions step by step and perform a complete calculus analysis including critical points, extrema, increasing and decreasing intervals, concavity, inflection points, asymptotes, variation tables, interactive graphs, tangent lines, and downloadable PDF reports.

Analyze a Function

Enter a real-valued function of x. Use notation such as x^3, sin(x), sqrt(x), exp(x), and log(x).

Function f(x)

Variable

Graph / analysis from

Derivative order

Detect and respect the natural domain of the function

Analyzing function...

Quick Analysis

Key results from the complete function analysis.

Domain

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First Derivative

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Second Derivative

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Symmetry

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Critical Points

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Local Maximum

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Local Minimum

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Inflection Points

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First Derivative

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Step-by-Step Differentiation

Derivative steps will appear here.

Second & Higher Derivatives

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Roots, Critical Points & Extrema

Analyze intercepts and derivative sign changes to determine maxima and minima.

Roots / X-Intercepts

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Y-Intercept

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Critical Points

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Extrema

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Increasing & Decreasing Intervals

The sign of \(f'(x)\) determines where the function rises and falls.

Increasing

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Decreasing

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Interval	Sign of f′(x)	Behavior
Analysis will appear here.

Variation Table

A complete visual summary of critical values, derivative signs, and function behavior.

The variation table will appear after analysis.

Concavity & Inflection Points

Concavity is determined using the sign of the second derivative.

Concave Up

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Concave Down

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Inflection Points

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Concavity table will appear here.

A solution of \(f''(x)=0\) is only an inflection-point candidate. The calculator verifies that the concavity actually changes before reporting an inflection point.

Limits & Asymptotes

Important Limits

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Asymptotes

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Function Sign Analysis

Determine where the function is positive, negative, or equal to zero.

f(x) > 0

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f(x) < 0

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Sign table will appear here.

Interactive Function & Derivative Graph

Compare the original function with its first and second derivatives and visualize important calculus features.

f(x) f′(x) f″(x) Critical points Inflection points Asymptotes

Tangent & Normal Line Calculator

Choose a point \(x=a\) to calculate the tangent and normal lines to the function.

x = a

Point

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Slope

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Tangent Line

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Normal Line

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Download Complete Function Analysis

Generate a professional MathVisualLab PDF report containing the function, derivatives, step-by-step work, critical points, extrema, variation table, concavity, inflection points, asymptotes, graph, and tangent analysis.

How Function Analysis Works

First Derivative

\[ f'(x)>0 \Rightarrow f(x)\text{ is increasing} \] \[ f'(x)<0 \Rightarrow f(x)\text{ is decreasing} \]

Second Derivative

\[ f''(x)>0 \Rightarrow f(x)\text{ is concave up} \] \[ f''(x)<0 \Rightarrow f(x)\text{ is concave down} \]

Symbolic results are preferred whenever they can be established reliably. Numerical searches are limited to the selected analysis interval and will be clearly identified as numerical approximations rather than exact global results.