University:
Bismarck State College
Course:
MATH 146 | Applied Calculus I
Academic year:

2019
Views:

468

Pages:

3
Author:

Alexa Livingston

Sin (x) sqrt(1 - x ^ 2) Cos(x) - 1/(sqrt(1 - x ^ 2)) tan (0 1/(1 + x ^ 2) Cof (x) - 1+x2 arcsec(x) x * sqrt(x ^ 2 - 1) CSC(x) Sinh160 sqrt(1 + x ^ 2) (20), 4007 1 - x ^ 2 * sin k < 1 Coth(x) 세기 X2-1 Sech(x) x * sqrt(1 - x ^ 2) CSC) X * sqrt(1 + X ^ 2) f(x) d/dx (f(x)) f(x) d/dx (f(x)) A 0 arcsin(x) 1 sqrt(1 - x ^ 2) sqrt(1 - x ^ 2) x ^ n n * x ^ (n - 1) arccos(x) -1 e ^ x e ^ x arctan(x) 1/(1 + x ^ 2) a ^ x a ^ x * ln(a) arccot(x) -1 ln(x) 1/x arcsec(x) 1 1 + x ^ 2 x * sqrt(x ^ 2 - 1) Loga(x) 1/x * log_a(e) arccsc(x) -1 x * sqrt(x ^ 2 - 1) sin(x) cos(x) sinh^(- 1) (x) 1/(sqrt(1 + x ^ 2)) cos(x) - sin(x) cosh^(- 1) (x) 1/(sqrt(x ^ 2 - 1)) tan(x) sec^2 (x) tanh^(- 1) (x) 1/(1 - x ^ 2) cot(x) csc^2 (x) coth^(- 1) (x) - 1/(x ^ 2 - 1) sec(x) sec(x) * tan(x) sec sech-1(x) overline x * sqrt(1 - x ^ 2) csc(x) - csc(x) * cot(x) csch-1(x) -1 x * sqrt(1 + x ^ 2) sin h * (x) cosh(x) coth(x) -csch²(x) cosh(x) sinh(x) sech(x) - sech(x) * tanh(x) tanh(x) sech²(x) csch(x) - csch(x) * coth(x) Proofs: 1- y = arc * sin x = arcsin(x) Then x = sin y Derivative both sides w.r.t x, then 1 = cos y * d/dx (y) -> d/dx (y) = 1/(cos y) = 1/(sqrt(1 - sin^2 y)) = 1/(sqrt(1 - x ^ 2)) cos^2 y + sin^2 y = 1 2- y = arc * tan x = arctan(x) Then x = tan y Derivative both sides w.r.t x, then 1 + tan^2 y = sec^2 y 1 = sec^2 y * d/dx (y) -> d/dx (y) = 1/(sec^2 y) = 1/(1 + tan^2 y) = 1/(1 + x ^ 2) 3- y = arc * sec(x) = arcsec(x) Then x = sec(y) Derivative both sides w.r.t x, then = 1/(x * sqrt(x ^ 2 - 1)) 1 = sec(y) * tan y * d/dx (y) -> d/dx (y) = 1/(sec(y) * tan y) = 1/(sec(y) * sqrt(sec^2 y - 1)) 1 + tan^2 y = sec^2 y