You entered:

ln(sin(2x))x^5x-1

What you mean?

log(sin(2*x))*x^5*x - 1

log(sin(2*x))*x^(5*x) - 1

log(sin(2*x))*x^(5*x - 1)

log(sin(2*x))*x^(5*x - 1)

Derivative of ln(sin(2x))x^5x-1

The solution

You have entered [src]

               5      
log(sin(2*x))*x *x - 1

$$x x^{5} \log{\left(\sin{\left(2 x \right)} \right)} - 1$$

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Detail solution

Differentiate $x x^{5} \log{\left(\sin{\left(2 x \right)} \right)} - 1$ term by term:
1. Apply the product rule:
  
  $\frac{d}{d x} f{\left(x \right)} g{\left(x \right)} = f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}$
  
  $f{\left(x \right)} = x^{5} \log{\left(\sin{\left(2 x \right)} \right)}$ ; to find $\frac{d}{d x} f{\left(x \right)}$ :
  1. Apply the product rule:
    
    $\frac{d}{d x} f{\left(x \right)} g{\left(x \right)} = f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}$
    
    $f{\left(x \right)} = \log{\left(\sin{\left(2 x \right)} \right)}$ ; to find $\frac{d}{d x} f{\left(x \right)}$ :
    1. Let $u = \sin{\left(2 x \right)}$ .
    2. The derivative of $\log{\left(u \right)}$ is $\frac{1}{u}$ .
    3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \sin{\left(2 x \right)}$ :
      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 $\frac{d}{d x} 2 x$ :
        
        The derivative of a constant times a function is the constant times the derivative of the function.
        
        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 of the chain rule is:
      
      $\frac{2 \cos{\left(2 x \right)}}{\sin{\left(2 x \right)}}$
    $g{\left(x \right)} = x^{5}$ ; to find $\frac{d}{d x} g{\left(x \right)}$ :
    1. Apply the power rule: $x^{5}$ goes to $5 x^{4}$
    The result is: $\frac{2 x^{5} \cos{\left(2 x \right)}}{\sin{\left(2 x \right)}} + 5 x^{4} \log{\left(\sin{\left(2 x \right)} \right)}$
  $g{\left(x \right)} = x$ ; to find $\frac{d}{d x} g{\left(x \right)}$ :
  1. Apply the power rule: $x$ goes to $1$
  The result is: $x^{5} \log{\left(\sin{\left(2 x \right)} \right)} + x \left(\frac{2 x^{5} \cos{\left(2 x \right)}}{\sin{\left(2 x \right)}} + 5 x^{4} \log{\left(\sin{\left(2 x \right)} \right)}\right)$
2. The derivative of the constant $-1$ is zero.
The result is: $x^{5} \log{\left(\sin{\left(2 x \right)} \right)} + x \left(\frac{2 x^{5} \cos{\left(2 x \right)}}{\sin{\left(2 x \right)}} + 5 x^{4} \log{\left(\sin{\left(2 x \right)} \right)}\right)$
Now simplify:

$2 x^{5} \left(\frac{x}{\tan{\left(2 x \right)}} + 3 \log{\left(\sin{\left(2 x \right)} \right)}\right)$

The answer is:

$2 x^{5} \left(\frac{x}{\tan{\left(2 x \right)}} + 3 \log{\left(\sin{\left(2 x \right)} \right)}\right)$

The graph

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The first derivative [src]

  /                        5         \                   
  |   4                 2*x *cos(2*x)|                  5
x*|5*x *log(sin(2*x)) + -------------| + log(sin(2*x))*x 
  \                        sin(2*x)  /

$$x^{5} \log{\left(\sin{\left(2 x \right)} \right)} + x \left(\frac{2 x^{5} \cos{\left(2 x \right)}}{\sin{\left(2 x \right)}} + 5 x^{4} \log{\left(\sin{\left(2 x \right)} \right)}\right)$$

Simplify

The second derivative [src]

     /                               2    2                     \
   4 |     2                      2*x *cos (2*x)   12*x*cos(2*x)|
2*x *|- 2*x  + 15*log(sin(2*x)) - -------------- + -------------|
     |                                 2              sin(2*x)  |
     \                              sin (2*x)                   /

$$2 x^{4} \left(- 2 x^{2} - \frac{2 x^{2} \cos^{2}{\left(2 x \right)}}{\sin^{2}{\left(2 x \right)}} + \frac{12 x \cos{\left(2 x \right)}}{\sin{\left(2 x \right)}} + 15 \log{\left(\sin{\left(2 x \right)} \right)}\right)$$

Simplify

The third derivative [src]

     /                                 2    2           3    3           3                         \
   3 |      2                      18*x *cos (2*x)   4*x *cos (2*x)   4*x *cos(2*x)   45*x*cos(2*x)|
4*x *|- 18*x  + 30*log(sin(2*x)) - --------------- + -------------- + ------------- + -------------|
     |                                   2                3              sin(2*x)        sin(2*x)  |
     \                                sin (2*x)        sin (2*x)                                   /

$$4 x^{3} \left(\frac{4 x^{3} \cos{\left(2 x \right)}}{\sin{\left(2 x \right)}} + \frac{4 x^{3} \cos^{3}{\left(2 x \right)}}{\sin^{3}{\left(2 x \right)}} - 18 x^{2} - \frac{18 x^{2} \cos^{2}{\left(2 x \right)}}{\sin^{2}{\left(2 x \right)}} + \frac{45 x \cos{\left(2 x \right)}}{\sin{\left(2 x \right)}} + 30 \log{\left(\sin{\left(2 x \right)} \right)}\right)$$

Simplify

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Derivative of ln(sin(2x))x^5x-1

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