We'll recall the identity that gives the cosecant
function:
csc x = 1/sin x
y =
10csc(3x/2) = 10/sin(3x/2)
We'll calculate the 1st
derivative of the function using the quotient rule:
dy/dx =
[(10)'*sin(3x/2) - 10*[sin(3x/2)]'/[sin(3x/2)]^2
dy/dx = -
15*[cos(3x/2)]/[sin(3x/2)]^2
Now, we'll calculate the value
of the first derivative at x = 2 /6
dy/dx = -
15*[cos(6 /12)]/[sin(6
/12)]^2
dy/dx = -15*cos(
/2)/(sin /2)^2
But cos /2 = 0, therefore dy/dx =
0 at x = 2 /6.
Therefore the function has an extreme at
x = 2 /6.
The requested value of the 1st
derivative, at x = 2 /6 is dy/dx = 0.
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