Derivative Calculator

Possibly neither. Simplification isn't unique: 2sin(x)cos(x) and sin(2x) are the same function, as are forms differing by factored-out constants or combined logarithms. Before assuming an error, test equivalence — evaluate both forms at two or three x values (the evaluate-at-a-point field here does one side); matching outputs means matching functions. Genuine mismatches usually trace to input typos: a missing parenthesis changes e^(x+1) into e^x + 1.

Standard math notation with x as the variable: ^ for powers, sin/cos/tan and their arc- inverses, e^x, ln for natural log, log for base-10, sqrt, and abs. Implicit multiplication is handled — 2x, 3sin(x), and x(x+2) all parse. The main syntax trap is exponent grouping: e^2x parses as (e²)·x, so write e^(2x) when the whole thing is the exponent. When in doubt, over-parenthesize; the simplifier cleans up.

The rate of change of the rate of change — concretely: f′ says whether f is rising or falling, f″ says whether it's curving up (concave up, f″ > 0, shaped like a cup) or down (concave down, like a frown). That powers the second-derivative test: at a point where f′ = 0, positive f″ means a local minimum, negative means a maximum. In physics terms, if f is position, f′ is velocity and f″ is acceleration.

It shows each derivative order and the simplified result, but not the rule-by-rule derivation (product rule applied here, chain rule there) — symbolic engines apply rules simultaneously in ways that don't map to clean textbook steps. The honest workflow: do the derivation by hand, use this to verify the destination. If your answer disagrees, the error is usually in one specific rule application, and knowing the correct target makes it findable.