Mathematics · Algebra & Calculus · Differential Calculus
Derivative Calculator
Computes the derivative of standard polynomial, trigonometric, exponential, and logarithmic functions using fundamental differentiation rules, and evaluates the derivative at a specified point.
Calculator
Formula
These are the core differentiation rules: the Power Rule states that the derivative of x raised to the power n equals n times x to the power (n-1); the Exponential Rule gives the derivative of e^(ax) as a·e^(ax); the Natural Log Rule gives the derivative of ln(ax) as 1/x; the Sine Rule gives the derivative of sin(ax) as a·cos(ax); and the Cosine Rule gives the derivative of cos(ax) as -a·sin(ax). In each case, a is a constant coefficient, n is the exponent, and x is the variable. The derivative f'(x) describes the instantaneous rate of change of the function f(x) at any point x.
Source: Stewart, J. (2015). Calculus: Early Transcendentals (8th ed.). Cengage Learning. Chapters 2–3.
How it works
A derivative, denoted f'(x) or df/dx, measures how rapidly a function changes with respect to its variable at any given point. Geometrically, the derivative equals the slope of the tangent line drawn to the function's graph at that point. If the derivative is positive, the function is increasing at that point; if negative, it is decreasing; if zero, the function has a local extremum (maximum or minimum). Derivatives are the foundation of differential calculus, introduced independently by Isaac Newton and Gottfried Wilhelm Leibniz in the 17th century.
This calculator applies five core differentiation rules. The Power Rule states that if f(x) = a·xⁿ, then f'(x) = a·n·xⁿ⁻¹ — each term's exponent steps down by one and multiplies the coefficient. The Exponential Rule gives f'(x) = a·e^(ax) for f(x) = e^(ax), reflecting the unique property that exponential functions are their own derivatives scaled by the inner coefficient. The Natural Logarithm Rule gives f'(x) = 1/x for f(x) = ln(ax), because the constant a cancels in differentiation. The Sine Rule gives f'(x) = a·cos(ax), and the Cosine Rule gives f'(x) = −a·sin(ax), capturing the cyclic relationship between trigonometric functions under differentiation.
Practical applications of derivatives span virtually every quantitative discipline. In physics, velocity is the derivative of position with respect to time, and acceleration is the derivative of velocity. In economics, marginal cost and marginal revenue are derivatives of total cost and revenue functions. In machine learning, gradient descent algorithms use derivatives to iteratively minimize loss functions and train neural networks. Civil and mechanical engineers use derivatives to analyze stress distributions, beam deflections, and fluid flow rates.
Worked example
Suppose you want to find the derivative of the function f(x) = 3x² and evaluate it at x = 4.
Step 1 — Identify the function type: This is a power function with coefficient a = 3 and exponent n = 2.
Step 2 — Apply the Power Rule: f'(x) = a·n·x^(n−1) = 3 × 2 × x^(2−1) = 6x.
Step 3 — Evaluate at x = 4: f'(4) = 6 × 4 = 24.
Interpretation: At x = 4, the function f(x) = 3x² is increasing at a rate of 24 units per unit of x. The tangent line to the curve at the point (4, 48) has a slope of 24.
Now consider a trigonometric example: f(x) = sin(2x) evaluated at x = π/6.
Step 1: Coefficient a = 2, function type: Sine.
Step 2 — Apply the Sine Rule: f'(x) = a·cos(ax) = 2·cos(2x).
Step 3 — Evaluate: f'(π/6) = 2·cos(2 × π/6) = 2·cos(π/3) = 2 × 0.5 = 1.0.
The slope of sin(2x) at x = π/6 (approximately 0.5236) is exactly 1, meaning the function is rising steeply at that point.
Limitations & notes
This calculator handles single-term functions of one variable using the five most fundamental differentiation rules. It does not support composite functions requiring the Chain Rule (e.g., sin(x²)), products or quotients requiring the Product or Quotient Rule (e.g., x²·sin(x)), implicit differentiation, partial derivatives of multivariable functions, or higher-order derivatives (second, third derivatives). For natural logarithm functions, the input ax must be strictly positive — the logarithm is undefined for non-positive arguments, and the calculator will return NaN if ax ≤ 0. Very large values of the exponent n or the coefficient a in exponential functions may produce numerical overflow (Infinity) due to floating-point limitations in JavaScript. For complex symbolic differentiation, dedicated computer algebra systems such as Wolfram Alpha, Mathematica, or SymPy are more appropriate tools.
Frequently asked questions
What is the Power Rule for derivatives?
The Power Rule states that if f(x) = a·xⁿ, then f'(x) = a·n·xⁿ⁻¹. For example, the derivative of 5x³ is 5 × 3 × x² = 15x². It is the most widely used differentiation rule and applies to any real-valued exponent n, including fractions and negative numbers.
Why is the derivative of e^x equal to itself?
The natural exponential function e^x is unique because it is its own derivative: d/dx[e^x] = e^x. This property stems from the definition of e as the base for which the exponential function grows at a rate exactly equal to its current value. When the exponent includes a constant coefficient a, as in e^(ax), the Chain Rule introduces a factor of a, giving a·e^(ax).
What does it mean when the derivative equals zero?
When f'(x) = 0 at a point, the tangent line to the curve is horizontal, indicating a critical point. This is a necessary condition for a local maximum or minimum — the so-called First Derivative Test. However, it is not sufficient on its own; you must check the Second Derivative or the sign change of f'(x) around that point to classify it as a maximum, minimum, or saddle point.
Can I use this calculator for the Chain Rule?
No. This calculator handles single-term functions like a·xⁿ, e^(ax), ln(ax), sin(ax), and cos(ax) where a is a constant. The Chain Rule is needed for composite functions like sin(x²) or e^(x³). For those cases, a symbolic differentiation tool such as Wolfram Alpha or a CAS (Computer Algebra System) should be used instead.
How are derivatives used in optimization problems?
Optimization problems seek to find maximum or minimum values of a function. Setting the derivative f'(x) = 0 identifies critical points where these extrema may occur. Engineers use this to minimize material costs, physicists use it to find equilibrium positions, and economists use it to maximize profit or minimize cost functions. The Second Derivative Test — checking whether f''(x) is positive (minimum) or negative (maximum) at the critical point — then confirms the nature of each extremum.
Last updated: 2025-01-15 · Formula verified against primary sources.