Calculus

Calculus Solver

Enter a function f(x) to find its derivative with respect to x.

d/dx
Examples:

Differentiation Rules

Power Rule
d/dx [x^n] = nx^(n-1)
Product Rule
[fg]' = f'g + fg'
Chain Rule
d/dx [f(g(x))] = f'(g(x))g'(x)
Constant Rule
d/dx [c] = 0

How to Differentiate Functions Step by Step

Differentiation is the process of finding the derivative of a function — a fundamental operation in calculus that reveals how a quantity changes with respect to another. Whether you're analyzing the velocity of a moving object, the slope of a curve, or the rate of growth of an investment, derivatives provide the mathematical framework. TheCalcPro's calculus solver automates the differentiation process while showing every intermediate step, so you can learn the method and verify homework solutions simultaneously.

Essential Differentiation Rules

Power Rule: d/dx [xⁿ] = n·xⁿ⁻¹

Product Rule: d/dx [u·v] = 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)

These four rules, combined with the derivatives of elementary functions (d/dx sin x = cos x, d/dx eˣ = eˣ, d/dx ln x = 1/x), are sufficient to differentiate virtually any expression encountered in undergraduate mathematics.

Step-by-Step Example: Differentiate f(x) = 3x⁴ − 2x² + 7

  1. Apply the power rule to each term individually (the derivative of a sum is the sum of derivatives).
  2. Term 1: d/dx [3x⁴] = 3 · 4x³ = 12x³
  3. Term 2: d/dx [−2x²] = −2 · 2x¹ = −4x
  4. Term 3: d/dx [7] = 0 (the derivative of any constant is zero)
  5. Combine: f′(x) = 12x³ − 4x

The derivative tells us the instantaneous rate of change of f at any point x. For instance, at x = 2: f′(2) = 12(8) − 4(2) = 96 − 8 = 88, meaning the function is increasing rapidly at that point.

Practical Applications

Derivatives are used in physics to compute velocity and acceleration from position functions, in economics to analyze marginal cost and revenue, in machine learning for gradient descent optimization, and in engineering for control systems. To graph your functions and their derivatives visually, try our Function Plotter. For algebraic equation solving, visit the Algebra Solver.

Calculus Solver FAQ

A derivative measures the instantaneous rate of change of a function with respect to its variable. Geometrically, it represents the slope of the tangent line to the function's graph at any point. It is formally defined as f′(x) = lim(h→0) [f(x+h) − f(x)] / h. Derivatives are fundamental to physics (velocity is the derivative of position), economics (marginal cost), and engineering (rate of change analysis).

The power rule states that if f(x) = xⁿ, then f′(x) = n·xⁿ⁻¹. You bring the exponent down as a coefficient and reduce the exponent by 1. For example, the derivative of x⁵ is 5x⁴, and the derivative of x⁻² is −2x⁻³. This rule works for any real exponent n, including fractions and negative numbers.

The chain rule is used when differentiating a composite function — a function nested inside another function. It states: d/dx [f(g(x))] = f′(g(x)) · g′(x). For example, to differentiate sin(x³), the outer function is sin(u) with derivative cos(u), and the inner function is u = x³ with derivative 3x². The result is 3x² · cos(x³).

The product rule differentiates a product of two functions: d/dx [u · v] = u′v + uv′. The quotient rule differentiates a ratio: d/dx [u/v] = (u′v − uv′) / v². A useful tip: the quotient rule can be derived from the product rule by writing u/v as u · v⁻¹ and applying the chain rule to v⁻¹.