Enter a power function f(x) = a·xⁿ and a point x₀ to get the power-rule derivative f'(x₀), a numerical central-difference cross-check, and a two-sided numerical limit estimate as x approaches x₀.
Derivatives and limits are the two ideas calculus is built from: a limit describes what a function approaches, and a derivative describes how fast it changes. This calculator works with power functions f(x) = a·xⁿ — the building block of every polynomial — and shows you the exact power-rule derivative alongside a numerical cross-check, so you can see both methods agree.
For f(x) = a·xⁿ, the derivative is f'(x) = a·n·xⁿ⁻¹ — multiply by the exponent, then reduce the exponent by one. It is the single most-used differentiation rule because every polynomial term has this form, and it extends directly to negative and fractional exponents (so it also covers 1/x = x⁻¹ and √x = x^0.5).
The definition of a derivative is a limit: f'(x₀) = lim(h→0) [f(x₀+h) − f(x₀−h)] / 2h. This calculator evaluates that expression at a very small h = 10⁻⁴ instead of taking h all the way to zero — a central difference approximation. It should land extremely close to the exact power-rule answer, which is a useful way to build confidence that you've applied the power rule correctly, and it's exactly the technique numerical software uses when no exact formula is available.
A two-sided limit exists at x₀ only if the function approaches the same value from the left (x₀⁻) and from the right (x₀⁺). For a smooth power function away from any singularity (like x₀ = 0 with a negative exponent), both sides converge to f(x₀), which is exactly what "continuous at x₀" means. This calculator evaluates f just below and just above x₀ so you can see the left- and right-hand approach values directly, not just a limit stated as fact.
Take f(x) = 3x² and x₀ = 4. The power rule gives f'(x) = 3·2·x¹ = 6x, so f'(4) = 24. The central-difference check with h = 10⁻⁴ gives (3(4.0001)² − 3(3.9999)²) / 0.0002 ≈ 24.0000, matching the exact answer. f(4) = 3(16) = 48, and the left/right limit evaluations both land at essentially 48, confirming the function is continuous there.
Floating-point arithmetic and the finite step size h introduce a tiny truncation error — typically far smaller than the displayed precision. For smooth functions like power functions, a central difference with h = 10⁻⁴ typically agrees with the exact derivative to at least 6-7 significant figures.
f(x) = a·xⁿ with n < 0 is undefined at x = 0 (division by zero), so f(0), the derivative, and the limit all show as undefined. This correctly reflects that functions like 1/x have a vertical asymptote at x = 0 — the two-sided limit doesn't exist there because the function diverges to +∞ on one side and −∞ on the other.
Yes — the power rule f'(x) = a·n·xⁿ⁻¹ holds for any real exponent n, including fractions. For example, f(x) = x^0.5 (the square root function) has f'(x) = 0.5·x⁻⁰·⁵, which is defined for x > 0.
A limit describes the value a function approaches as its input approaches some point — it's a statement about the function's behavior near a point. A derivative is itself defined as a specific limit (the limit of the average rate of change as the interval shrinks to zero), so every derivative is built from a limit, but not every limit describes a derivative.