Question #920f7

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Answer 1 · 2017-08-10T01:55:58

$\frac{d}{d x} [- {csc}^{2} x] = 2 cot x {csc}^{2} x$ $d/(dx) [-csc^2x] = color(blue)(2cotxcsc^2x$

We're asked to find the derivative

$\frac{d}{d x} [- {csc}^{2} x]$ $d/(dx) [-csc^2x]$

Let's first use the chain rule:

$\frac{d}{d x} [- {csc}^{2} x] = - \frac{d}{d u} [u^{2}] \frac{d u}{d x}$ $d/(dx) [-csc^2x] = -d/(du) [u^2] (du)/(dx)$

where

$= - 2 csc x \frac{d}{d x} [csc x]$ $= -2cscxd/(dx)[cscx]$

The derivative of $csc x$ $cscx$ is $- cot x csc x$ $-cotxcscx$ :

$= - 2 csc x (- cot x csc x)$ $= -2cscx(-cotxcscx)$

Or

$= color(blue)(ulbar(|stackrel(" ")(" "2cotxcsc^2x" ")|)$