Compute the derivative f'(x) and evaluate at a point.
Supported: sin, cos, tan, exp, log, sqrt, abs, ^, pi, e, *
The derivative f'(x) = lim(h→0) [f(x+h)−f(x)]/h. For polynomials: d/dx[xⁿ] = nxⁿ⁻¹. For common functions: d/dx[sin(x)] = cos(x), d/dx[eˣ] = eˣ, d/dx[ln(x)] = 1/x.
The derivative represents the instantaneous rate of change (slope of the tangent line) at any point. In physics, the derivative of position is velocity, and the derivative of velocity is acceleration.
The second derivative f''(x) is the derivative of the derivative. It tells us the concavity: f'' > 0 means concave up (minimum), f'' < 0 means concave down (maximum). It's also acceleration in physics.
The chain rule: d/dx[f(g(x))] = f'(g(x)) × g'(x). It's used to differentiate composite functions. For example, d/dx[sin(x²)] = cos(x²) × 2x.