Math Final

f(x) = sin x

f’(x) = cos x

f(x) = cos x

f’(x) = -sin x

f(x) = tan x

f’(x) = sec ² x

f(x) = sec x

f’(x) = sec x tan x

f(x) = cot x

f’(x) = -csc ² x

f(x) = csc x

f’(x) = -csc x cot x

f(x) = arcsin x

f’(x) = 1/rad(1-x²)

f(x) = arccos x

f’(x) = -1/rad(1-x²)

f(x) = arctan x

f’(x) = 1/x²+1

f(x) = arccsc x

f’(x) = -1/|x|*rad(x²-1)

f(x) = arcsec x

f’(x) = 1/ |x| *rad(x²-1)

f(x) = arccot x

f’(x) = -1/x²+1

f(x) = a^u

f'(x) = a^u * ln(a) * u'

f(x) = log a u

f'(x) = 1/(u ln(a)) * u'

f(x) = e^u

f’(x) = e^u * u’

Non Differentiabilites

Cusp, Corner, Vertical Tan Line, Discontinuity

Continuous

Two sided limit are equal and both equal to f(c)

Differentiable

The function is continuous, and two sided derivatives match (at that point)

Diff implies continuity, does not go other way,

implies local linearity

[f^-1(x)]’ = 1/f’(f^-1(x))

if f is continuous on domain, f^-1 is, if diff on interval containing c and f’( c) is non zero, inverse is also diff at that point

Horizontal tan line

set top of derivative to 0

Vertical tan line

set bottom of derivative to 0 (DNE)