AI Derivative Solver Powered by Gemini

Calculate derivatives with detailed steps, visualize functions and their derivatives, and understand calculus concepts with our AI-powered derivative calculator.

Calculate Derivative Learn More

d/dx [f(x)]

Enter your function to get:

Symbolic derivatives
Step-by-step solution
Graph visualization
Critical points

Derivative Calculator

Enter a function f(x) to calculate its derivative

f(x) =

Try these functions:

x^2 sin(x) e^x ln(x) x^3 - 4*x 1/x sqrt(x) tan(x)

Key Features

Step-by-Step Solutions

Get comprehensive, detailed explanations of derivative calculations using various rules like power rule, chain rule, and product rule.

Interactive Graphing

Visualize functions and their derivatives side by side. Explore the relationship between a function and its rate of change.

Critical Point Analysis

Automatically identify and analyze critical points, inflection points, and other key features of functions to understand their behavior.

Frequently Asked Questions

What is a derivative?

A derivative represents the rate of change of a function with respect to a variable. It tells you how quickly a function is changing at a specific point. For a function f(x), the derivative f'(x) or df/dx gives the slope of the tangent line to the function at any point x.

What are the basic derivative rules?

The basic derivative rules include:

  • Power Rule: d/dx(xⁿ) = n·xⁿ⁻¹
  • Constant Rule: d/dx(c) = 0
  • Sum Rule: d/dx(f(x) + g(x)) = f'(x) + g'(x)
  • Product Rule: d/dx(f(x)·g(x)) = f'(x)·g(x) + f(x)·g'(x)
  • Quotient Rule: d/dx(f(x)/g(x)) = (f'(x)·g(x) - f(x)·g'(x))/[g(x)]²
  • Chain Rule: d/dx(f(g(x))) = f'(g(x))·g'(x)

What are critical points?

Critical points are points where the derivative of a function equals zero or is undefined. They are important because they include potential locations of local maxima, local minima, and inflection points. To determine the type of critical point, you can use the second derivative test:

  • If f'(x) = 0 and f''(x) > 0, the point is a local minimum
  • If f'(x) = 0 and f''(x) < 0, the point is a local maximum
  • If f'(x) = 0 and f''(x) = 0, further testing is needed (inflection point or higher-order critical point)

What are some applications of derivatives?

Derivatives have numerous applications in science, engineering, economics, and other fields:

  • Physics: Calculating velocity and acceleration from position functions
  • Economics: Marginal cost, revenue, and profit analysis
  • Optimization: Finding maximum or minimum values of functions
  • Engineering: Analyzing rates of change in systems
  • Biology: Modeling population growth rates
  • Machine Learning: Gradient descent optimization algorithms

What functions can this solver handle?

Our derivative calculator can handle a wide range of functions, including:

  • Polynomial functions (e.g., x³ + 2x² - 5x + 3)
  • Trigonometric functions (e.g., sin(x), cos(x), tan(x))
  • Exponential functions (e.g., e^x, 2^x)
  • Logarithmic functions (e.g., ln(x), log₁₀(x))
  • Combinations of these functions using addition, subtraction, multiplication, division, and composition

The solver uses advanced techniques to handle complex expressions and provide accurate step-by-step solutions.

How to Use This Tool

1

Enter Your Function

Input your function using standard mathematical notation. For example, type "x^2 + 3*x - 5" for x² + 3x - 5. Use "*" for multiplication and "^" for exponents.

2

Select Derivative Order

Choose the order of the derivative you want to calculate (1st derivative, 2nd derivative, etc.). You can also enter a specific point to evaluate the derivative at that value.

3

Calculate and Explore

Click "Calculate Derivative" to see the results. The summary tab shows the derivative formula and evaluation. Switch to the graph tab to visualize the function and its derivatives.

4

Understand the Process

View the detailed step-by-step solution to understand how the derivative was calculated. The explanation includes all relevant calculus rules and simplification steps.

Input Format Tips

  • Use "sin(x)", "cos(x)", "tan(x)" for trigonometric functions
  • Use "e^x" or "exp(x)" for the exponential function
  • Use "ln(x)" for natural logarithm
  • Use "sqrt(x)" for square root
  • Use parentheses to group expressions, e.g., "sin(x^2)"