Question #52b72

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Answer 1 · 2018-02-14T08:18:11

By using the common derivative arccot formula and the chain rule, we get $\frac{α}{α^{2} + x^{2}}$ $alpha/(alpha^2+x^2)$

For the given problem:

$y =$ $y=$ arccot $(\frac{α}{x})$ $(alpha/x)$

We must apply a common derivative formula of:
$\frac{d}{d x} ($ $d/dx($ arccot $(u)$ $(u)$ ) $= - \frac{1}{u^{2} + 1}$ $=-1/(u^2+1)$

and the chain rule (this is only because our "u" is something other than just x)

in our problem we assign the following for ease:
$f = ($ $f=($ arccot $(u))$ $(u))$
$u = \frac{α}{x}$ $u=alpha/x$
our derivative will be:
$= \frac{d}{d u} ($ $=d/(du)($ arccot $(u)$ $(u)$ ) $\cdot \frac{d}{d x} (\frac{α}{x})$ $*d/dx(alpha/x)$
We can then calculate them separately and simplify:
$\frac{d}{d u} ($ $d/(du)($ arccot $(u)$ $(u)$ ) = $- \frac{1}{(\frac{α}{x}) + 1}$ $color(red)(-1/((alpha/x)+1))$
$\frac{d}{d x} (\frac{α}{x}) = α \cdot \frac{d}{d x} \frac{1}{x} = α \cdot \frac{d}{d x} x^{- 1} = - \frac{α}{x^{2}}$ $d/(dx)(alpha/x)=alpha*d/(dx)1/x=alpha*d/(dx)x^-1=color(blue)(-alpha/x^2)$
Multiply together and simplify
$- \frac{1}{(\frac{α}{x}) + 1}$ $color(red)(-1/((alpha/x)+1))$ $\cdot$ $*$ $- \frac{α}{x^{2}}$ $color(blue)(-alpha/x^2)$

For a final answer of:
$= \frac{α}{α^{2} + x^{2}}$ $=alpha/(alpha^2+x^2)$