Derivative Calculator

How to Use the Derivative Calculator

1. Select function type: Choose from power functions, polynomials, trigonometric functions (sin, cos, tan), exponential functions, logarithms, or product rule applications.
2. Enter parameters: Input the required values based on the selected function type. For power functions, enter the exponent. For polynomials, enter all coefficients.
3. View the derivative: The calculator instantly displays both the original function and its derivative using standard mathematical notation.
4. Understand the rule: See which differentiation rule was applied (power rule, sum rule, product rule, etc.) to help you learn the process.
5. Try examples: Click the quick example buttons to see common derivatives and understand how different functions behave under differentiation.

What is a Derivative?

In calculus, the derivative measures how a function changes as its input changes. Geometrically, the derivative at a point represents the slope of the tangent line to the function's graph at that point. The derivative f'(x) of a function f(x) gives the instantaneous rate of change, which has countless applications in physics, engineering, economics, and optimization problems.

The derivative is fundamental to calculus and appears everywhere in science and engineering. Velocity is the derivative of position with respect to time. Acceleration is the derivative of velocity. In economics, marginal cost is the derivative of the total cost function. Understanding derivatives is essential for optimization, motion analysis, and modeling dynamic systems.

Differentiation Rules

Power Rule: For f(x) = x^n, the derivative is f'(x) = nx^(n-1). This is the most fundamental rule and works for any real number n. Example: The derivative of x³ is 3x².

Constant Multiple Rule: The derivative of cf(x) is c·f'(x), where c is a constant. Constants factor out of derivatives. Example: The derivative of 5x² is 5·2x = 10x.

Sum Rule: The derivative of f(x) + g(x) is f'(x) + g'(x). Derivatives distribute over addition and subtraction. Example: The derivative of x³ + x² is 3x² + 2x.

Product Rule: For the product f(x)·g(x), the derivative is f'(x)·g(x) + f(x)·g'(x). This rule is essential when differentiating products of functions.

Common Function Derivatives

Trigonometric functions: d/dx[sin(x)] = cos(x), d/dx[cos(x)] = -sin(x), d/dx[tan(x)] = sec²(x). These patterns appear frequently in physics and engineering, especially in wave analysis and oscillatory motion.

Exponential function: d/dx[e^x] = e^x. The exponential function is its own derivative, a unique and powerful property used in growth models, differential equations, and probability.

Natural logarithm: d/dx[ln(x)] = 1/x. The logarithm's derivative appears in integration, solving differential equations, and analyzing logarithmic scales.

Constant: The derivative of any constant is zero, since constants don't change.

Applications of Derivatives

Derivatives find critical points where functions reach maximum or minimum values, essential for optimization in business, engineering, and science. Setting f'(x) = 0 locates these points. The second derivative test (using f''(x)) classifies critical points as maxima, minima, or saddle points.

In physics, derivatives describe motion: velocity is the derivative of position, and acceleration is the derivative of velocity. In economics, marginal analysis uses derivatives to study how small changes in production affect costs and revenues. In machine learning, gradient descent algorithms use derivatives to optimize neural networks.

Understanding Derivative Notation

Prime notation: f'(x) represents the derivative of f(x). The second derivative is f''(x).

Leibniz notation: dy/dx or df/dx emphasizes the rate of change interpretation and is useful in applied contexts.

Operator notation: d/dx is the differentiation operator. For example, d/dx[x²] = 2x.

Higher derivatives: The second derivative f''(x) measures how the rate of change itself changes (acceleration if f is position). Third and higher derivatives have specialized applications.

Integration is the inverse of differentiation - use the Integral Calculator to find antiderivatives. The Limit Calculator helps understand the foundations of derivatives.