trig identities and elementary series

d/dx(tan x)

sec²(x)

d/dx(cot x)

-csc²(x)

d/dx (sec x)

sec x tan x

d/dx (csc x)

-csc x cot x

d/dx (a^x)

a^x ln a

d/dx (log_ax)

1/(x ln a)

d/dx (arcsin x)

1/sqrt(1 - x²)

d/dx(arccos x)

-1/sqrt(1 - x²)

d/dx (arctan x)

1/(1 + x²)

d/dx (arccotan x) (remember arccot is just negative arctan shifted up)

-1/1+x²

d/dx (arcsec x)

1/ (|x| sqrt(x² - 1))

d/dx (arccossec x)

-1/(|x| sqrt(x² - 1))

taylor series for 1/x

(-1)ⁿ (x-1)ⁿ

TAYLOR SERIES

f(c) + f’(c)(x-c) + (f’’(c)/2!)(x-c)² + … + (f ⁿ(c)/n!)(x-c)ⁿ

mcclaurin series

taylor series centered at 0

taylor series for 1/x+1

(-1)ⁿ xⁿ

taylor series for ln(x)

((-1)^n-1 (x-1)ⁿ) / n

taylor series for e^x

(xⁿ) / n!

taylor series for cos x

((-1)ⁿ x²ⁿ) / (2n)!

taylor series for sin x

((-1)ⁿ x²ⁿ⁺¹) / (2n+1)!

pythagorean identities (there’s 4 things here)

sin²x + cos²x = 1, 1 - cos²x = sin²x, 1 + tan²x = sec²x, 1 + cot²x = csc²x

double angle formulas (1 for sin, 3 for cos)

sin(2x) = 2 sinx cosx, cosx = cos²x - sin²x, cosx = 2cos²x -1, cosx = 1 - 2sin²x

power reduction for sin²x

(1 - cos(2x)) / 2

power reduction for cos²x

(1 + cos(2x)) / 2

inverse identity thing (if h is the inverse of f)

h’(x) = 1 / f’(h(x))