Question

10 POINTS!!! FULL ANSWER WITH FULL STEP BY STEP SOLUTION PLEASE

Answer 1

s = d(sec-1(x))/dx = 1/x*sqrt(x^2 - 1)

c = d(csc-1(x))/dx = - 1/sqrt(x^4 - x^2)

Answer = s - c
Answer = 1/x*sqrt(x^2 - 1) + 1/sqrt(x^4 - x^2)
Answer = 1/x sqrt(x^2 - 1) + 1/x*sqrt(x^2 - 1)
Answer = 1/x sqrt(x^2 - 1) * [1 + 1]
Answer = 1/2*x*sqrt(x^2 - 1)
f'(x) = 2/[x sqrt(x^2 - 1)]

Now you are ready to put in values for f'(x)
f'(4) = 2/[ 4 sqrt(16 - 1)]
f'(4) = 2/[ 4 sqrt(15)
f'(4) = 1/2 * sqrt(15)/15
f'(4) = sqrt(15)/30

f'(-1) = 2/(-1) sqrt( ((-1)^2 - 1)
Undefined. I don't know how you've been instructed to handle 1/0. I call it undefined.

f'(1/2) = 2/((1/2)sqrt(1/2)^2 - 1) another undefined.
f'(1/2) = 4/(sqrt(1/4 - 1)
f'(1/2) = 4/sqrt(-3/4)
f'(1/2) = 8/ i*sqrt(3)
f'(1/2) = 8* sqrt(3) / 3*i
f'(1/2) = 8*i* sqrt(3)/3*i^2
f'(1/2) = - 8i sqrt(3) / 3

Answer 2

f'(x) = (d/dx)(arcsec(x) -arccsc(x)) = 1/(x^2*√(1 -1/x^2)) -(-1)/(x^2*√(1 -1/x^2))

a) f'(x) = (d/dx)(arcsec(x) -arccsc(x)) = 2/√(x^4 -x^2)

b) f'(4) = 2/√(4^4 -4^2) = 2/√240 = (√15)/30

c) f'(-1) = 2/√((-1)^4 -(-1)^2) = 2/0 = undefined

d) f'(1/2) = 2/√((1/2)^4 -(1/2)^2) = 2/√(-3/16) = 8/√-3 = -i(8/3)√3