Chapter 2_Derivatives of basic functions

Derivatives of basic elementary functions (xα )′ = αxα−1 c′ = c-constant, x′ = √ ′ ( x) = √ x α = 1, α = 12 , ( )′ 1 =− x x (sin x)′ = cos x (cos x)′ = − sin x (tan x)′ = (cot x)′ = − α = −1 cos2 x sin2 x (ax )′ = ax ln a (ex )′ = ex (loga x)′ = (ln x)′ = a > 0, a ̸= 1 x ln a a > 0, a ̸= 1 x 10 (arcsin x)′ = √ − x2 11 (arccos x)′ = − √ − x2 12 (arctan x)′ = 13 (arccot x)′ = − 14 (sinh x)′ = cosh x 15 (cosh x)′ = sinh x 1 + x2 1 + x2 coth2 x 16 (tanh x)′ = 17 (coth x)′ = − sinh2 x Rules of differentiation Given two differentiable functions u = u(x), v = v(x) [u(x) ± v(x)]′ = u′ (x) ± v ′ (x); [u(x)v(x)]′ = u′ (x)v(x) + u(x)v ′ (x); If c is a constant then [c · u(x)]′ = cu′ (x) [ ]′ u(x) u′ (x)v(x) − u(x)v ′ (x) = ; v(x) v (x) [ ]′ v ′ (x) =− v(x) v (x) The derivative of composite function y = f [φ(x)] y ′ = f ′ [φ(x)] φ′ (x) ...1 coth2 x 16 (tanh x)′ = 17 (coth x)′ = − sinh2 x Rules of differentiation Given two differentiable functions u = u(x), v = v(x) [u(x) ± v(x)]′ = u′ (x) ± v ′ (x); [u(x)v(x)]′... (x) [ ]′ u(x) u′ (x)v(x) − u(x)v ′ (x) = ; v(x) v (x) [ ]′ v ′ (x) =− v(x) v (x) The derivative of composite function y = f [φ(x)] y ′ = f ′ [φ(x)] φ′ (x)