Derivative Calculator: Compute Derivatives Step by Step

Derivatives are a cornerstone of calculus, measuring the rate at which a function changes. Whether you are a student learning calculus, an engineer analyzing systems, or a researcher modeling data, our Derivative Calculator computes derivatives of mathematical functions and shows the step-by-step solution using standard differentiation rules.

What Is a Derivative?

The derivative of a function at a point represents the slope of the tangent line at that point. Geometrically, it measures how steep the function is. Physically, it represents the rate of change: velocity is the derivative of position with respect to time, acceleration is the derivative of velocity, and marginal cost is the derivative of total cost with respect to quantity.

The calculator supports differentiation rules including: power rule (d/dx x^n = nx^(n-1)), product rule (d/dx (uv) = u'v + uv'), quotient rule (d/dx (u/v) = (u'v - uv')/v²), chain rule (d/dx f(g(x)) = f'(g(x)) × g'(x)), and trigonometric, exponential, and logarithmic derivatives.

Using the Derivative Calculator

Enter your function using standard mathematical notation. Use x as the variable. Supported operations include +, -, *, /, ^ (power), sqrt(), sin(), cos(), tan(), log(), ln(), exp(). The calculator shows the derivative and a step-by-step breakdown of each differentiation rule applied.

You can also evaluate the derivative at a specific point by entering an x value. The calculator computes the instantaneous rate of change at that point and shows the equation of the tangent line.

Differentiation Rules

Power Rule

The most commonly used rule: d/dx (x^n) = n × x^(n-1). For example, d/dx (x^5) = 5x^4, and d/dx (1/x) = d/dx (x^(-1)) = -x^(-2) = -1/x².

Product Rule

Used when two functions are multiplied: d/dx (f(x) × g(x)) = f'(x)g(x) + f(x)g'(x). Example: d/dx (x² × sin(x)) = 2x sin(x) + x² cos(x).

Chain Rule

Used for composed functions: d/dx (f(g(x))) = f'(g(x)) × g'(x). Example: d/dx (sin(x²)) = cos(x²) × 2x. The chain rule is essential for differentiating any function inside another function.

Trigonometric Derivatives

d/dx sin(x) = cos(x), d/dx cos(x) = -sin(x), d/dx tan(x) = sec²(x), d/dx sec(x) = sec(x)tan(x).

Applications of Derivatives

• Optimization: Find maximum and minimum values by setting derivative to zero. Used in business for profit maximization and cost minimization.
• Physics: Velocity and acceleration are derivatives of position. Force is the derivative of potential energy.
• Economics: Marginal cost, marginal revenue, and elasticity of demand are all derivative-based concepts.
• Engineering: Rates of change in electrical circuits, fluid dynamics, and structural analysis all rely on derivatives.
• Machine learning: Gradient descent, the core optimization algorithm in neural networks, uses derivatives to minimize loss functions.

Real-World Example

A company's profit function is P(x) = -2x² + 100x - 500, where x is the number of units produced:

• Derivative: P'(x) = -4x + 100 (marginal profit)
• Maximum profit occurs at P'(x) = 0: -4x + 100 = 0, x = 25 units
• Maximum profit: P(25) = -2(25)² + 100(25) - 500 = -1,250 + 2,500 - 500 = $750
• At x=25, derivative P'(25) = 0 — each additional unit neither increases nor decreases profit

If the company currently produces 20 units, P'(20) = 20 > 0, meaning increasing production by 1 unit increases profit by approximately $20.

Start Calculating

Use our Derivative Calculator below to compute derivatives and understand rates of change. Also check our Integral Calculator for integration and our Graphing Calculator for visualizing functions.