How do you find the derivative of $y = {csc}^{- 1} (\frac{x}{2})$ $y=csc^-1 (x/2)$ ?

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Answer 1 · 2018-01-03T09:30:36

$\frac{d y}{d x} = - \frac{2}{x \sqrt{x^{2} - 4}}$ $(dy)/(dx)=-2/(xsqrt(x^2-4))$

As $y = {csc}^{- 1} (\frac{x}{2})$ $y=csc^(-1)(x/2)$

$\frac{x}{2} = csc y$ $x/2=cscy$

taking derivative on both sides

$\frac{1}{2} = - cot y csc y \cdot \frac{d y}{d x}$ $1/2=-cotycscy*(dy)/(dx)$

or $\frac{d y}{d x} = - \frac{1}{2} \cdot \frac{1}{cot y} \cdot \frac{1}{csc y}$ $(dy)/(dx)=-1/2*1/coty*1/cscy$

= $- \frac{1}{2} \cdot \frac{1}{\sqrt{{csc}^{2} y - 1}} \cdot \frac{1}{csc y}$ $-1/2*1/sqrt(csc^2y-1)*1/cscy$

= $- \frac{1}{2} \cdot \frac{1}{\sqrt{\frac{x^{2}}{4} - 1}} \cdot \frac{2}{x}$ $-1/2*1/sqrt(x^2/4-1)*2/x$

= $- \frac{2}{x \sqrt{x^{2} - 4}}$ $-2/(xsqrt(x^2-4))$

How do you find the derivative of y=csc−1(x2)y=csc^-1 (x/2)?