Calculus of a Single Variable, Chapter 2, 2.3, Section 2.3, Problem 59

y=(1+csc(x))/(1-csc(x))
differentiating by applying the quotient rule,
y'=((1-csc(x))d/dx(1+csc(x))-(1+csc(x))d/dx(1-csc(x)))/((1-csc(x))^2)
y'=((1-csc(x))(-csc(x)cot(x))-(1+csc(x))(csc(x)cot(x)))/(1-csc(x))^2
y'=(csc(x)cot(x)(-1+csc(x)-1-csc(x)))/(1-csc(x))^2
y'=(-2csc(x)cot(x))/(1-csc(x))^2
Now let us evaluate the derivative at x=pi/6
We know that csc(pi/6)=2 and cot(pi/6)=
Plug in the above values in y',
y'(pi/6)=(-2*2*sqrt(3))/(1-2)^2
y'(pi/6)=-4sqrt(3)=-6.9282
Graph of the derivative is attached.