Derivative of log3(x)*sin2x

The solution

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log(x)         
------*sin(2*x)
log(3)

$$\frac{\log{\left(x \right)}}{\log{\left(3 \right)}} \sin{\left(2 x \right)}$$

(log(x)/log(3))*sin(2*x)

Detail solution

Apply the quotient rule, which is:

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

$f{\left(x \right)} = \log{\left(x \right)} \sin{\left(2 x \right)}$ and $g{\left(x \right)} = \log{\left(3 \right)}$ .

To find $\frac{d}{d x} f{\left(x \right)}$ :
1. Apply the product rule:
  
  $\frac{d}{d x} f{\left(x \right)} g{\left(x \right)} = f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}$
  
  $f{\left(x \right)} = \log{\left(x \right)}$ ; to find $\frac{d}{d x} f{\left(x \right)}$ :
  1. The derivative of $\log{\left(x \right)}$ is $\frac{1}{x}$ .
  $g{\left(x \right)} = \sin{\left(2 x \right)}$ ; to find $\frac{d}{d x} g{\left(x \right)}$ :
  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:
    
    $2 \cos{\left(2 x \right)}$
  The result is: $2 \log{\left(x \right)} \cos{\left(2 x \right)} + \frac{\sin{\left(2 x \right)}}{x}$
To find $\frac{d}{d x} g{\left(x \right)}$ :
1. The derivative of the constant $\log{\left(3 \right)}$ is zero.
Now plug in to the quotient rule:

$\frac{2 \log{\left(x \right)} \cos{\left(2 x \right)} + \frac{\sin{\left(2 x \right)}}{x}}{\log{\left(3 \right)}}$
Now simplify:

$\frac{2 x \log{\left(x \right)} \cos{\left(2 x \right)} + \sin{\left(2 x \right)}}{x \log{\left(3 \right)}}$

The answer is:

$\frac{2 x \log{\left(x \right)} \cos{\left(2 x \right)} + \sin{\left(2 x \right)}}{x \log{\left(3 \right)}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

sin(2*x)   2*cos(2*x)*log(x)
-------- + -----------------
x*log(3)         log(3)

$$\frac{2 \log{\left(x \right)} \cos{\left(2 x \right)}}{\log{\left(3 \right)}} + \frac{\sin{\left(2 x \right)}}{x \log{\left(3 \right)}}$$

Simplify

The second derivative [src]

  sin(2*x)                       4*cos(2*x)
- -------- - 4*log(x)*sin(2*x) + ----------
      2                              x     
     x                                     
-------------------------------------------
                   log(3)

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

Simplify

The third derivative [src]

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

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

Simplify

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Derivative of log3(x)*sin2x

The graph:

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The solution