Question

First derivative of √{cosec2x).show with full step.​

Answer 1

9514 1404 393

Answer:

  -cot(2x)√csc(2x)

Step-by-step explanation:

Using the chain rule, ...

  (d/dx)(√u) = u'/(2√u)

Here, we have u = csc(2x), so u' = -2cot(2x)csc(2x).

Then ...

  (d/dx)(√csc(2x)) = (-2cot(2x)csc(2x))/(2√csc(2x)) = -cot(2x)√csc(2x)

Answer 2

Answer:

$- \sf \displaystyle \: \frac{ \cos(2x) }{ \sin ^{2} (2x)\sqrt{ \csc(2x) } }$

Step-by-step explanation:

we are given a derivative

$\displaystyle \: \frac{d}{dx} ( \sqrt{ \csc(2x) } )$

and said to figure out the first derivative

to do so

recall chain rule:

$\sf\displaystyle \: \frac{d}{dx} (f(g(x)) = \frac{d}{dg} (f(g(x)) \times \frac{d}{dx} (g)$

so we get

$\displaystyle \: g(x) = \csc(2x)$

rewrite the derivative using the chain rule:

$\displaystyle \: \frac{d}{dg} ( \sqrt{ g } ) \times \frac{d}{dx} ( \csc(2x) )$

use square root derivative rule to simplify:

$\displaystyle \: \frac{1}{ 2\sqrt{g} } \times \frac{d}{dx} ( \csc(2x) )$

now we need to again use chain rule composite function derivative to simplify

where we'll take a new function n so we won't mess up two g's and we'll take 2x as n

use composite function derivative to simplify:

$\sf \displaystyle \: \frac{1}{ 2\sqrt{g} } \times \frac{d}{dn}( \csc(n) ) \times \frac{d}{dx} (2x)$

use derivative formula to simplify derivatives:

$\sf \displaystyle \: \frac{1}{ 2\sqrt{g} } \times - \cot(n) \csc(n) \times 2$

substitute the value of n:

$\sf \displaystyle \: \frac{1}{ 2\sqrt{g} } \times - 2\cot(2x) \csc(2x)$

substitute the value of g:

$\sf \displaystyle \: \frac{1}{ 2\sqrt{ \csc(2x) } } \times - 2\cot(2x) \csc(2x)$

now we need our trigonometric skills to simplify

rewrite cot and csc:

$\sf \displaystyle \: \frac{1}{ 2\sqrt{ \csc(2x) } } \times - 2 \dfrac{ \cos(2x) }{ \sin(2x) } \dfrac{1}{ \sin(2x) }$

simplify multiplication:

$\sf \displaystyle \: \frac{1}{ \cancel{ \: 2}\sqrt{ \csc(2x) } } \times \cancel{- 2} \dfrac{ \cos(2x) }{ \sin ^{2} (2x) }$

simplify multiplication:

$- \sf \displaystyle \: \frac{ \cos(2x) }{ \sin ^{2} (2x)\sqrt{ \csc(2x) } }$