# Question c60ff

Feb 14, 2017

$\frac{d}{\mathrm{dx}} \left(\sqrt{\cot} x\right) = - {\csc}^{2} \frac{x}{2 \sqrt{\cot x}}$

#### Explanation:

$\frac{d}{\mathrm{dx}} \left(\sqrt{\cot} x\right) = {\lim}_{h \to 0} \frac{\sqrt{\cot} \left(x + h\right) - \sqrt{\cot x}}{h}$

Rationalize the numerator using the identity $\left({a}^{2} - {b}^{2}\right) = \left(a + b\right) \left(a - b\right)$

$\frac{d}{\mathrm{dx}} \left(\sqrt{\cot} x\right) = {\lim}_{h \to 0} \left(\frac{\sqrt{\cot} \left(x + h\right) - \sqrt{\cot x}}{h}\right) \left(\frac{\sqrt{\cot} \left(x + h\right) + \sqrt{\cot x}}{\sqrt{\cot} \left(x + h\right) + \sqrt{\cot x}}\right)$

d/dx (sqrt cot x ) = lim_(h->0) ( cot(x+h) - (cot x))/(h (sqrt cot(x+h) + sqrt(cot x))

Now use the trigonometric formulas for the sum of angles:

$\cot \left(x + h\right) = \cos \frac{x + h}{\sin} \left(x + h\right) = \frac{\cos x \cos h - \sin x \sin h}{\cos x \sin h + \sin x \cos h}$

divide numerator and denominator by $\sin x \sin h$

$\cot \left(x + h\right) = \frac{\cot x \cot h - 1}{\cot x + \cot h}$

end evaluate the difference:

$\cot \left(x + h\right) - \cot x = \frac{\cot x \cot h - 1}{\cot x + \cot h} - \cot x$

$\cot \left(x + h\right) - \cot x = \frac{\left(\cot x \cot h - 1\right) - \cot x \left(\cot x + \cot h\right)}{\cot x + \cot h}$

$\cot \left(x + h\right) - \cot x = \frac{\left(\cancel{\cot x \cot h} - 1 - {\cot}^{2} x - \cancel{\cot x \cot h}\right)}{\cot x + \cot h}$

$\cot \left(x + h\right) - \cot x = - \frac{1 + {\cot}^{2} x}{\cot x + \cot h}$

and as:

$1 + {\cot}^{2} x = 1 + {\cos}^{2} \frac{x}{\sin} ^ 2 x = \frac{{\sin}^{2} x + {\cos}^{2} x}{\sin} ^ 2 x = \frac{1}{\sin} ^ 2 x = {\csc}^{2} x$

we have finally:

$\cot \left(x + h\right) - \cot x = - {\csc}^{2} \frac{x}{\cot x + \cot h}$

So:

d/dx (sqrt cot x ) = lim_(h->0) ( - csc^2x/(cotx +cot h ))1/(h (sqrt cot(x+h) + sqrt(cot x))#

$\frac{d}{\mathrm{dx}} \left(\sqrt{\cot} x\right) = {\lim}_{h \to 0} \left(- {\csc}^{2} \frac{x}{\sqrt{\cot} \left(x + h\right) + \sqrt{\cot x}}\right) \left(\frac{1}{h \left(\cot x + \coth\right)}\right)$

Now note that:

${\lim}_{h \to 0} \frac{1}{h \left(\cot x + \coth\right)} = {\lim}_{h \to 0} \frac{1}{h \cot x + \frac{\cosh}{\frac{\sinh}{h}}} = 1$

and we can conclude:

$\frac{d}{\mathrm{dx}} \left(\sqrt{\cot} x\right) = - {\csc}^{2} \frac{x}{2 \sqrt{\cot x}}$