how to take the derivative of a natural log function.

Question

so the function given is a(t)=ln((1-cost)/(1+cost))^4

 

my teacher said I can't take the natural log of a natural log so I don't know how to take the derivative of this function. please help me.

Answer 1

a(t) = ln[(1-cost)/(1+cost)]⁴

 

       = 4{ln[(1-cost)/(1+cost)]}

 

       = 4{ln(1-cost) - ln(1+cost)}

 

a'(t) = 4{(1-cost)'/(1-cost) - (1+cost)'/(1+cost)}

 

       = 4{sint/(1-cost) + sint/(1+cost)}

 

       = 4{[sint(1+cost) + sint(1-cost)]/[(1-cost)(1+cost)]}

 

       = 4{2sint/(1-cos²t}

 

       = 8sint/sin²t = 8/sint = 8csct