Derivative Calculator

What Is a Derivative?

A derivative measures how fast a function changes at any given point. If a function describes the position of a car over time, the derivative gives the car's speed at each moment. If the function describes the total cost of producing x items, the derivative gives the marginal cost per additional item. Mathematically, the derivative of f(x) is the limit of (f(x+h) - f(x)) / h as h approaches zero. Enter any function in the calculator above to find its derivative instantly with step-by-step explanation.

Basic Derivative Rules

Power rule: The derivative of x^n is n times x^(n-1). The derivative of x³ is 3x². The derivative of x^5 is 5x^4. Constant rule: The derivative of any constant is 0. Constant multiple: The derivative of c times f(x) is c times f'(x). The derivative of 5x³ is 15x². Sum/difference: Differentiate each term separately. The derivative of x³ + 2x² - 5x is 3x² + 4x - 5. These rules handle most polynomial derivatives by inspection.

Chain Rule, Product Rule, and Quotient Rule

For more complex functions, three additional rules apply. Product rule: The derivative of f(x) times g(x) is f'(x)g(x) + f(x)g'(x). For x² times sin(x): 2x sin(x) + x² cos(x). Quotient rule: The derivative of f(x)/g(x) is (f'g - fg') / g². Chain rule: The derivative of f(g(x)) is f'(g(x)) times g'(x). For sin(3x): cos(3x) times 3 = 3cos(3x). The chain rule handles composite functions (functions inside functions) and is the most frequently used advanced rule in calculus.

Common Derivatives to Know

Trigonometric: derivative of sin(x) = cos(x). Derivative of cos(x) = -sin(x). Derivative of tan(x) = sec²(x). Exponential: derivative of e^x = e^x (the only function equal to its own derivative). Derivative of a^x = a^x times ln(a). Logarithmic: derivative of ln(x) = 1/x. Derivative of log(x) = 1/(x times ln(10)). These fundamental derivatives combine with the rules above to differentiate virtually any function you encounter in calculus courses and applications.

What Does the Derivative Tell You?

The derivative has geometric and practical interpretations. Geometrically, f'(a) is the slope of the tangent line to f(x) at x = a. A positive derivative means the function is increasing. A negative derivative means it is decreasing. A zero derivative means the function has a horizontal tangent, potentially a maximum, minimum, or inflection point. Practically, derivatives describe rates: speed (derivative of position), acceleration (derivative of speed), marginal cost (derivative of total cost), and growth rate (derivative of population). Finding where the derivative equals zero identifies optimal points, which is why calculus is essential for optimization in engineering, economics, and science.

Applications of Derivatives

Physics uses derivatives everywhere: velocity is the derivative of position, acceleration is the derivative of velocity, force is the derivative of potential energy. Economics uses marginal analysis (derivatives of cost, revenue, and profit functions) to make production decisions. Machine learning uses gradients (multi-variable derivatives) to train neural networks through backpropagation. Medical pharmacology models drug concentration decay rates. Engineers find maximum stress points in structures. Any field that needs to optimize, predict rates of change, or understand how one quantity responds to changes in another relies on derivatives as the fundamental mathematical tool.

Higher-Order Derivatives

The second derivative f''(x) is the derivative of the derivative. It measures the rate of change of the rate of change. In physics, the second derivative of position is acceleration. In economics, the second derivative of a cost function indicates whether marginal costs are increasing or decreasing. The second derivative also determines concavity: if f''(x) is positive, the graph curves upward (concave up). If negative, it curves downward (concave down). Points where concavity changes (f'' = 0) are inflection points. Higher derivatives (third, fourth, etc.) exist and appear in Taylor series, vibration analysis, and advanced physics.

Implicit Differentiation

When y is not isolated (like in x² + y² = 25, the equation of a circle), implicit differentiation finds dy/dx by differentiating both sides with respect to x and solving for dy/dx. For x² + y² = 25: 2x + 2y(dy/dx) = 0, so dy/dx = -x/y. This technique handles curves that cannot be written as y = f(x) and is essential for related rates problems in calculus, where multiple changing quantities are connected by an equation.

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