#separator:tab #html:false #tags column:4 Constant: d/dx [c] 0 math-derivative-rules Power rule: d/dx [xⁿ] n xⁿ⁻¹ math-derivative-rules Constant multiple rule d/dx [c f(x)] = c f'(x) math-derivative-rules Sum rule d/dx [f + g] = f' + g' math-derivative-rules Difference rule d/dx [f − g] = f' − g' math-derivative-rules Product rule d/dx [fg] = f'g + fg' math-derivative-rules Quotient rule d/dx [f/g] = (f'g − fg') / g² math-derivative-rules Chain rule d/dx [f(g(x))] = f'(g(x)) · g'(x) math-derivative-rules d/dx [x] 1 math-derivative-rules d/dx [x²] 2x math-derivative-rules d/dx [x³] 3x² math-derivative-rules d/dx [√x] 1/(2√x) math-derivative-rules d/dx [1/x] −1/x² math-derivative-rules d/dx [eˣ] eˣ math-derivative-rules d/dx [aˣ] aˣ ln a math-derivative-rules d/dx [ln x] 1/x math-derivative-rules d/dx [log_a x] 1/(x ln a) math-derivative-rules d/dx [sin x] cos x math-derivative-rules d/dx [cos x] −sin x math-derivative-rules d/dx [tan x] sec² x math-derivative-rules d/dx [cot x] −csc² x math-derivative-rules d/dx [sec x] sec x tan x math-derivative-rules d/dx [csc x] −csc x cot x math-derivative-rules d/dx [arcsin x] 1/√(1 − x²) math-derivative-rules d/dx [arccos x] −1/√(1 − x²) math-derivative-rules d/dx [arctan x] 1/(1 + x²) math-derivative-rules d/dx [sinh x] cosh x math-derivative-rules d/dx [cosh x] sinh x math-derivative-rules d/dx [e^(2x)] 2e^(2x) math-derivative-rules d/dx [sin(3x)] 3 cos(3x) math-derivative-rules d/dx [(x² + 1)³] 6x(x² + 1)² math-derivative-rules d/dx [x eˣ] eˣ(x + 1) math-derivative-rules d/dx [x sin x] sin x + x cos x math-derivative-rules d/dx [ln(x²)] 2/x math-derivative-rules d/dx [5x⁴ − 3x + 7] 20x³ − 3 math-derivative-rules Derivative as slope The slope of the tangent line at a point math-derivative-rules Limit definition of derivative f'(x) = lim (h→0) [f(x + h) − f(x)] / h math-derivative-rules Second derivative The derivative of the derivative; tells concavity math-derivative-rules Critical point A point where f' = 0 or f' does not exist math-derivative-rules First derivative test f' changes from + to − at a local maximum, and − to + at a local minimum math-derivative-rules Second derivative test f'' > 0 gives a local minimum; f'' < 0 gives a local maximum math-derivative-rules Inflection point A point where concavity changes math-derivative-rules Concave up f'' > 0; the graph curves like a cup math-derivative-rules Concave down f'' < 0; the graph curves like a cap math-derivative-rules Velocity from position v(t) = s'(t) math-derivative-rules Acceleration from position a(t) = s''(t) math-derivative-rules Mean value theorem A smooth curve has a point where the slope equals the average slope over the interval math-derivative-rules L'Hôpital's rule For 0/0 or ∞/∞ forms, lim f/g = lim f'/g' (if that limit exists) math-derivative-rules Implicit differentiation Differentiate both sides with respect to x, using dy/dx for y terms math-derivative-rules Related rates Differentiate an equation in time to link rates of change math-derivative-rules Tangent line equation y − f(a) = f'(a)(x − a) math-derivative-rules Rolle's theorem If f(a) = f(b) and f is smooth, there is a point between where f' = 0 math-derivative-rules