Derivative Calculator – Find Function Derivatives
Calculate derivatives of mathematical functions symbolically
Table of Contents
How to Use
- Enter your mathematical function (e.g., x^2, sin(x), e^x)
- Specify the variable to differentiate with respect to (usually x)
- Click calculate to see the derivative
- Review the result showing the original function and its derivative
What is a Derivative?
A derivative represents the rate of change of a function with respect to a variable. It measures how a function's output changes as its input changes, providing the slope of the tangent line at any point on the function's graph.
Derivatives are fundamental in calculus and have applications in physics (velocity, acceleration), economics (marginal cost, marginal revenue), and many other fields.
Basic Differentiation Rules
- Power Rule: d/dx(x^n) = n·x^(n-1)
- Constant Rule: d/dx(c) = 0
- Sum Rule: d/dx(f + g) = f' + g'
- Product Rule: d/dx(f·g) = f'·g + f·g'
- Chain Rule: d/dx(f(g(x))) = f'(g(x))·g'(x)
Common Function Derivatives
| Function | Derivative |
|---|---|
| x^n | n·x^(n-1) |
| sin(x) | cos(x) |
| cos(x) | -sin(x) |
| tan(x) | sec²(x) |
| e^x | e^x |
| ln(x) | 1/x |
Applications of Derivatives
- Finding maximum and minimum values of functions
- Calculating velocity and acceleration in physics
- Optimizing business processes and costs
- Determining rates of change in natural phenomena
- Analyzing function behavior and graphing curves
Frequently Asked Questions
- What is the power rule for derivatives?
- The power rule states that the derivative of x^n is n·x^(n-1). For example, the derivative of x^3 is 3x^2, and the derivative of x^2 is 2x.
- What is the derivative of a constant?
- The derivative of any constant is always zero. Since constants don't change, their rate of change is zero.
- How do I find the derivative of trigonometric functions?
- Common trigonometric derivatives include: d/dx(sin(x)) = cos(x), d/dx(cos(x)) = -sin(x), and d/dx(tan(x)) = sec²(x).
- What is the chain rule?
- The chain rule is used to differentiate composite functions. It states that d/dx(f(g(x))) = f'(g(x))·g'(x). You differentiate the outer function and multiply by the derivative of the inner function.