Derivative of y=sin2x/lnx

The solution

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

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

sin(2*x)/log(x)

Detail solution

Apply the quotient rule, which is:

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

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

To find $\frac{d}{d x} f{\left(x \right)}$ :
1. Let $u = 2 x$ .
2. The derivative of sine is cosine:
  
  $\frac{d}{d u} \sin{\left(u \right)} = \cos{\left(u \right)}$
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:
  
  $2 \cos{\left(2 x \right)}$
To find $\frac{d}{d x} g{\left(x \right)}$ :
1. The derivative of $\log{\left(x \right)}$ is $\frac{1}{x}$ .
Now plug in to the quotient rule:

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

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

The answer is:

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

The graph

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Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

                           /      2   \         
                           |1 + ------|*sin(2*x)
              4*cos(2*x)   \    log(x)/         
-4*sin(2*x) - ---------- + ---------------------
               x*log(x)           2             
                                 x *log(x)      
------------------------------------------------
                     log(x)

$$\frac{- 4 \sin{\left(2 x \right)} - \frac{4 \cos{\left(2 x \right)}}{x \log{\left(x \right)}} + \frac{\left(1 + \frac{2}{\log{\left(x \right)}}\right) \sin{\left(2 x \right)}}{x^{2} \log{\left(x \right)}}}{\log{\left(x \right)}}$$

Simplify

The third derivative [src]

  /                           /      3         3   \                                   \
  |                           |1 + ------ + -------|*sin(2*x)     /      2   \         |
  |                           |    log(x)      2   |            3*|1 + ------|*cos(2*x)|
  |              6*sin(2*x)   \             log (x)/              \    log(x)/         |
2*|-4*cos(2*x) + ---------- - ------------------------------- + -----------------------|
  |               x*log(x)                3                             2              |
  \                                      x *log(x)                     x *log(x)       /
----------------------------------------------------------------------------------------
                                         log(x)

$$\frac{2 \left(- 4 \cos{\left(2 x \right)} + \frac{6 \sin{\left(2 x \right)}}{x \log{\left(x \right)}} + \frac{3 \left(1 + \frac{2}{\log{\left(x \right)}}\right) \cos{\left(2 x \right)}}{x^{2} \log{\left(x \right)}} - \frac{\left(1 + \frac{3}{\log{\left(x \right)}} + \frac{3}{\log{\left(x \right)}^{2}}\right) \sin{\left(2 x \right)}}{x^{3} \log{\left(x \right)}}\right)}{\log{\left(x \right)}}$$

Simplify

The graph

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Derivative of y=sin2x/lnx

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The solution