Calculus Question

Question

Find the derivative and simplify:

 

y = 2csc³(sqrtx)

Help!?

Answer 1

Rewrite the function using rational exponents.

 

y = 2csc³(x^1/2)

 

Let     u = x^1/2

 

Then,

 

y = 2csc³(u)

 

Using the chain rule,

 

y' = 2u' * -3csc²(u) * csc(u)cot(u)

 

y' = -2(1/2)x^-1/2  * 3csc²(x^1/2) * csc(x^1/2)cot(x^1/2)

 

y' = -3x^-1/2(1 / sin³(x^1/2))(cos(x^1/2) / sin(x^1/2))

 

y' = -3x^-1/2cos(x^1/2) / sin⁴(x^1/2)

 

y' = -3x^-1/2cot(x^1/2)csc³(x^1/2)