Derivative of y=(ln2x)/(sinx)

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log(2*x)
--------
 sin(x)

$$\frac{\log{\left(2 x \right)}}{\sin{\left(x \right)}}$$

d /log(2*x)\
--|--------|
dx\ sin(x) /

$$\frac{d}{d x} \frac{\log{\left(2 x \right)}}{\sin{\left(x \right)}}$$

Transform

Detail solution

Apply the quotient rule, which is:

$\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$

$f{\left(x \right)} = \log{\left(2 x \right)}$ and $g{\left(x \right)} = \sin{\left(x \right)}$ .

To find $\frac{d}{d x} f{\left(x \right)}$ :
1. Let $u = 2 x$ .
2. The derivative of $\log{\left(u \right)}$ is $\frac{1}{u}$ .
3. Then, apply the chain rule. Multiply by $\frac{d}{d x} 2 x$ :
  1. The derivative of a constant times a function is the constant times the derivative of the function.
    1. Apply the power rule: $x$ goes to $1$
    So, the result is: $2$
  The result of the chain rule is:
  
  $\frac{1}{x}$
To find $\frac{d}{d x} g{\left(x \right)}$ :
1. The derivative of sine is cosine:
  
  $\frac{d}{d x} \sin{\left(x \right)} = \cos{\left(x \right)}$
Now plug in to the quotient rule:

$\frac{- \log{\left(2 x \right)} \cos{\left(x \right)} + \frac{\sin{\left(x \right)}}{x}}{\sin^{2}{\left(x \right)}}$
Now simplify:

$\frac{- x \log{\left(2 x \right)} \cos{\left(x \right)} + \sin{\left(x \right)}}{x \sin^{2}{\left(x \right)}}$

The answer is:

$\frac{- x \log{\left(2 x \right)} \cos{\left(x \right)} + \sin{\left(x \right)}}{x \sin^{2}{\left(x \right)}}$

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The first derivative [src]

   1       cos(x)*log(2*x)
-------- - ---------------
x*sin(x)          2       
               sin (x)

$$- \frac{\log{\left(2 x \right)} \cos{\left(x \right)}}{\sin^{2}{\left(x \right)}} + \frac{1}{x \sin{\left(x \right)}}$$

Simplify

The second derivative [src]

       /         2   \                    
  1    |    2*cos (x)|            2*cos(x)
- -- + |1 + ---------|*log(2*x) - --------
   2   |        2    |            x*sin(x)
  x    \     sin (x) /                    
------------------------------------------
                  sin(x)

$$\frac{\left(1 + \frac{2 \cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}}\right) \log{\left(2 x \right)} - \frac{2 \cos{\left(x \right)}}{x \sin{\left(x \right)}} - \frac{1}{x^{2}}}{\sin{\left(x \right)}}$$

Simplify

The third derivative [src]

       /         2   \               /         2   \                
       |    2*cos (x)|               |    6*cos (x)|                
     3*|1 + ---------|               |5 + ---------|*cos(x)*log(2*x)
       |        2    |               |        2    |                
2      \     sin (x) /    3*cos(x)   \     sin (x) /                
-- + ----------------- + --------- - -------------------------------
 3           x            2                       sin(x)            
x                        x *sin(x)                                  
--------------------------------------------------------------------
                               sin(x)

$$\frac{- \frac{\left(5 + \frac{6 \cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}}\right) \log{\left(2 x \right)} \cos{\left(x \right)}}{\sin{\left(x \right)}} + \frac{3 \cdot \left(1 + \frac{2 \cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}}\right)}{x} + \frac{3 \cos{\left(x \right)}}{x^{2} \sin{\left(x \right)}} + \frac{2}{x^{3}}}{\sin{\left(x \right)}}$$

Simplify

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Derivative of y=(ln2x)/(sinx)

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