`y = (1 + csc(x))/(1 - csc(x)), (pi/6,-3)` Evaluate the derivative of the function at the given point. Use a graphing utility to verify your result.

Expert Answers
gsarora17 eNotes educator| Certified Educator

`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.

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