Find the derivative of the following functions: write the solution of each number

Question

4. y = Arcsec(sec 2x)

5. f(t)= Arcsin(cos t)

6. h(z) = sin (Arccos z)

Answer 1

Hello Kevin,

For your question:

a. y = Arcsec(sec(2x))

The derivative of arcsec() is uncommon to be seen. But for simplicity's sake,

d/dx(arcsec(x)) = 1/(abs(x)*sqrt(x^2-1))

So, with that in mind:

d/dx(y) = d/dx(arcsec(sec(2x)))

= 1/(abs(sec(2x))*sqrt((sec(2x))^2 -1))*d/dx(sec(2x))*d/dx(2x) through chain rule.

The rest I will leave for you to simplify.

b. y = Arcsin(cos(t))

d/dt(y) = 1/sqrt(1-(cos(t))^2)*d/dt(cos(t)) through chain rule.

I will leave the rest for you to simplify

c. h = sin(arccos(z))

d/dz(h) = d/dz(sin(arccos(z)))

= cos(arccos(z))*d/dz(arccos(z)), recall that d/dz(arccos(z)) = -1/sqrt(1-x^2)

The rest I will leave for you to simplify.

Hope this helps!