Single Variable Derivative Calculator

Welcome to the Single Variable Derivative Calculator, an advanced tool that computes derivatives of any single-variable function with detailed step-by-step solutions, differentiation rule identification, interactive graphing, and critical point analysis. Whether you are a calculus student learning differentiation, a teacher preparing examples, or an engineer solving rate-of-change problems, this calculator delivers accurate results with clear explanations.

What is a Derivative?

The derivative of a function measures the instantaneous rate of change of the function's output with respect to its input. Geometrically, the derivative at a point equals the slope of the tangent line to the function's graph at that point.

Differentiation Rules Reference

This calculator identifies which rules are applied at each step. Here is a quick reference:

How to Use This Calculator

1. Enter your function: Type the function using standard math notation. Use ^ for exponents, * for multiplication, and standard function names like sin(x), cos(x), e^x, ln(x), sqrt(x).
2. Set the variable: Usually x, but you can use any letter.
3. Choose the order: 1 for first derivative, 2 for second derivative, up to 10.
4. Evaluate at a point (optional): Enter a number or expression like pi to evaluate the derivative at that specific value.
5. Click "Compute Derivative": View the result, step-by-step solution, interactive graph, and critical points.

Supported Input Syntax

Understanding Higher-Order Derivatives

The second derivative \(f''(x)\) measures how the first derivative itself is changing — it tells you about the concavity of the original function. The third derivative measures the rate of change of the concavity (sometimes called "jerk" in physics). This calculator supports derivatives up to the 10th order, computing each one step by step.

Applications of Higher-Order Derivatives

• 2nd derivative: Concavity analysis, inflection points, acceleration in physics
• 3rd derivative: Jerk (rate of change of acceleration), curve sketching refinement
• 4th+ derivatives: Taylor series approximations, vibration analysis, signal processing

What Are Critical Points?

A critical point of a function is a value of \(x\) where the derivative equals zero or is undefined. At these points, the function may have a local maximum, local minimum, or inflection point. This calculator automatically solves \(f'(x) = 0\) and displays the critical points for your analysis.

Applications of Derivatives

• Physics: Velocity and acceleration from position functions
• Economics: Marginal cost, marginal revenue, and profit optimization
• Engineering: Rate of change analysis in control systems
• Biology: Population growth rate modeling
• Optimization: Finding maximum and minimum values of functions

Frequently Asked Questions

How do I enter a function into the derivative calculator?

Type the function using standard math notation. Use ^ or ** for exponents (x^3), * for multiplication (2*x), and standard function names like sin(x), cos(x), tan(x), e^x, ln(x), sqrt(x). The calculator automatically handles implicit multiplication like 2x.

What differentiation rules does this calculator show?

The calculator identifies and labels each differentiation rule used: Power Rule, Product Rule, Quotient Rule, Chain Rule, Sum/Difference Rule, Constant Multiple Rule, Exponential Rule, Trigonometric Rule, and Logarithmic Rule. Each step shows which rules were applied.

Can this calculator compute higher-order derivatives?

Yes, the calculator supports derivatives from 1st order through 10th order. Simply set the Order of Derivative field to the desired order. The step-by-step solution shows each successive differentiation.

What are critical points and why does the calculator show them?

Critical points are values of x where the derivative equals zero \(f'(x) = 0\). These points often correspond to local maxima, local minima, or inflection points of the original function. The calculator finds and displays these points to help you understand function behavior.

What functions are supported by this derivative calculator?

The calculator supports polynomials, trigonometric functions (sin, cos, tan, cot, sec, csc), inverse trigonometric functions (arcsin, arccos, arctan), exponential functions (e^x, a^x), logarithmic functions (ln, log), square roots (sqrt), absolute values, and compositions of these functions.

Additional Resources

• Derivative - Wikipedia
• Derivatives - Khan Academy