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Identical expressions
y=(lnsin2x+ four)^ three
y equally (ln sinus of 2x plus 4) cubed
y equally (ln sinus of 2x plus four) to the power of three
y=(lnsin2x+4)3
y=lnsin2x+43
y=(lnsin2x+4)³
y=(lnsin2x+4) to the power of 3
y=lnsin2x+4^3
Similar expressions
y=(lnsin2x-4)^3

The derivative of the function/ y=(lnsin2x+4)^3

Derivative of y=(lnsin2x+4)^3

The solution

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

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

(log(sin(2*x)) + 4)^3

Detail solution

Let $u = \log{\left(\sin{\left(2 x \right)} \right)} + 4$ .
Apply the power rule: $u^{3}$ goes to $3 u^{2}$
Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(\log{\left(\sin{\left(2 x \right)} \right)} + 4\right)$ :
1. Differentiate $\log{\left(\sin{\left(2 x \right)} \right)} + 4$ term by term:
  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 $\frac{d}{d x} 2 x$ :
      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:
    
    $\frac{2 \cos{\left(2 x \right)}}{\sin{\left(2 x \right)}}$
  4. The derivative of the constant $4$ is zero.
  The result is: $\frac{2 \cos{\left(2 x \right)}}{\sin{\left(2 x \right)}}$
The result of the chain rule is:

$\frac{6 \left(\log{\left(\sin{\left(2 x \right)} \right)} + 4\right)^{2} \cos{\left(2 x \right)}}{\sin{\left(2 x \right)}}$
Now simplify:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

   /                                                  2                           2    2             2                         \         
   |                         2                     cos (2*x)   (4 + log(sin(2*x))) *cos (2*x)   3*cos (2*x)*(4 + log(sin(2*x)))|         
48*|-12 + (4 + log(sin(2*x)))  - 3*log(sin(2*x)) + --------- + ------------------------------ - -------------------------------|*cos(2*x)
   |                                                  2                     2                                 2                |         
   \                                               sin (2*x)             sin (2*x)                         sin (2*x)           /         
-----------------------------------------------------------------------------------------------------------------------------------------
                                                                 sin(2*x)

$$\frac{48 \left(\left(\log{\left(\sin{\left(2 x \right)} \right)} + 4\right)^{2} + \frac{\left(\log{\left(\sin{\left(2 x \right)} \right)} + 4\right)^{2} \cos^{2}{\left(2 x \right)}}{\sin^{2}{\left(2 x \right)}} - \frac{3 \left(\log{\left(\sin{\left(2 x \right)} \right)} + 4\right) \cos^{2}{\left(2 x \right)}}{\sin^{2}{\left(2 x \right)}} - 3 \log{\left(\sin{\left(2 x \right)} \right)} - 12 + \frac{\cos^{2}{\left(2 x \right)}}{\sin^{2}{\left(2 x \right)}}\right) \cos{\left(2 x \right)}}{\sin{\left(2 x \right)}}$$

Simplify

The graph

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Derivative of y=(lnsin2x+4)^3

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The solution