Derivative of y=(sin2x)^cotgx

The solution

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   cot(a)         
sin      (2*x)*n*x

$$x n \sin^{\cot{\left(a \right)}}{\left(2 x \right)}$$

(sin(2*x)^cot(a)*n)*x

Detail solution

Apply the product rule:

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

$f{\left(x \right)} = n \sin^{\cot{\left(a \right)}}{\left(2 x \right)}$ ; to find $\frac{d}{d x} f{\left(x \right)}$ :
1. The derivative of a constant times a function is the constant times the derivative of the function.
  1. Let $u = \sin{\left(2 x \right)}$ .
  2. Apply the power rule: $u^{\cot{\left(a \right)}}$ goes to $\frac{u^{\cot{\left(a \right)}} \cot{\left(a \right)}}{u}$
  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 $\frac{d}{d x} 2 x$ :
      1. 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:
    
    $\frac{2 \sin^{\cot{\left(a \right)}}{\left(2 x \right)} \cos{\left(2 x \right)} \cot{\left(a \right)}}{\sin{\left(2 x \right)}}$
  So, the result is: $\frac{2 n \sin^{\cot{\left(a \right)}}{\left(2 x \right)} \cos{\left(2 x \right)} \cot{\left(a \right)}}{\sin{\left(2 x \right)}}$
$g{\left(x \right)} = x$ ; to find $\frac{d}{d x} g{\left(x \right)}$ :
1. Apply the power rule: $x$ goes to $1$
The result is: $\frac{2 n x \sin^{\cot{\left(a \right)}}{\left(2 x \right)} \cos{\left(2 x \right)} \cot{\left(a \right)}}{\sin{\left(2 x \right)}} + n \sin^{\cot{\left(a \right)}}{\left(2 x \right)}$
Now simplify:

$n \left(\frac{2 x}{\tan{\left(a \right)} \tan{\left(2 x \right)}} + 1\right) \sin^{\frac{1}{\tan{\left(a \right)}}}{\left(2 x \right)}$

The answer is:

$n \left(\frac{2 x}{\tan{\left(a \right)} \tan{\left(2 x \right)}} + 1\right) \sin^{\frac{1}{\tan{\left(a \right)}}}{\left(2 x \right)}$

The first derivative [src]

                            cot(a)                     
   cot(a)          2*n*x*sin      (2*x)*cos(2*x)*cot(a)
sin      (2*x)*n + ------------------------------------
                                 sin(2*x)

$$\frac{2 n x \sin^{\cot{\left(a \right)}}{\left(2 x \right)} \cos{\left(2 x \right)} \cot{\left(a \right)}}{\sin{\left(2 x \right)}} + n \sin^{\cot{\left(a \right)}}{\left(2 x \right)}$$

Simplify

The second derivative [src]

                   /             /       2           2            \\       
       cot(a)      |cos(2*x)     |    cos (2*x)   cos (2*x)*cot(a)||       
4*n*sin      (2*x)*|-------- - x*|1 + --------- - ----------------||*cot(a)
                   |sin(2*x)     |       2              2         ||       
                   \             \    sin (2*x)      sin (2*x)    //

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

Simplify

The third derivative [src]

                   /                                            /                    2           2         2           2            \         \       
                   |                                            |               2*cos (2*x)   cos (2*x)*cot (a)   3*cos (2*x)*cot(a)|         |       
                   |                                        2*x*|2 - 3*cot(a) + ----------- + ----------------- - ------------------|*cos(2*x)|       
                   |          2             2                   |                   2                2                   2          |         |       
       cot(a)      |     3*cos (2*x)   3*cos (2*x)*cot(a)       \                sin (2*x)        sin (2*x)           sin (2*x)     /         |       
4*n*sin      (2*x)*|-3 - ----------- + ------------------ + ----------------------------------------------------------------------------------|*cot(a)
                   |         2                2                                                  sin(2*x)                                     |       
                   \      sin (2*x)        sin (2*x)                                                                                          /

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

Simplify

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Derivative of y=(sin2x)^cotgx

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