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Derivative of x*ctgx
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Identical expressions
y=sin4^x/cos(8x)
y equally sinus of 4 to the power of x divide by co sinus of e of (8x)
y=sin4x/cos(8x)
y=sin4x/cos8x
y=sin4^x/cos8x
y=sin4^x divide by cos(8x)

The derivative of the function/ y=sin4^x/cos(8x)

Derivative of y=sin4^x/cos(8x)

The solution

You have entered [src]

   x    
sin (4) 
--------
cos(8*x)

$$\frac{\sin^{x}{\left(4 \right)}}{\cos{\left(8 x \right)}}$$

  /   x    \
d |sin (4) |
--|--------|
dx\cos(8*x)/

$$\frac{d}{d x} \frac{\sin^{x}{\left(4 \right)}}{\cos{\left(8 x \right)}}$$

Transform

Detail solution

Apply the quotient rule, which is:

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

$f{\left(x \right)} = \sin^{x}{\left(4 \right)}$ and $g{\left(x \right)} = \cos{\left(8 x \right)}$ .

To find $\frac{d}{d x} f{\left(x \right)}$ :
1. $\frac{d}{d x} \sin^{x}{\left(4 \right)} = \left(\log{\left(- \sin{\left(4 \right)} \right)} + i \pi\right) \sin^{x}{\left(4 \right)}$
To find $\frac{d}{d x} g{\left(x \right)}$ :
1. Let $u = 8 x$ .
2. The derivative of cosine is negative sine:
  
  $\frac{d}{d u} \cos{\left(u \right)} = - \sin{\left(u \right)}$
3. Then, apply the chain rule. Multiply by $\frac{d}{d x} 8 x$ :
  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: $8$
  The result of the chain rule is:
  
  $- 8 \sin{\left(8 x \right)}$
Now plug in to the quotient rule:

$\frac{8 \sin^{x}{\left(4 \right)} \sin{\left(8 x \right)} + \left(\log{\left(- \sin{\left(4 \right)} \right)} + i \pi\right) \sin^{x}{\left(4 \right)} \cos{\left(8 x \right)}}{\cos^{2}{\left(8 x \right)}}$
Now simplify:

$\frac{\left(8 \sin{\left(8 x \right)} + \left(\log{\left(- \sin{\left(4 \right)} \right)} + i \pi\right) \cos{\left(8 x \right)}\right) \sin^{x}{\left(4 \right)}}{\cos^{2}{\left(8 x \right)}}$

The answer is:

$\frac{\left(8 \sin{\left(8 x \right)} + \left(\log{\left(- \sin{\left(4 \right)} \right)} + i \pi\right) \cos{\left(8 x \right)}\right) \sin^{x}{\left(4 \right)}}{\cos^{2}{\left(8 x \right)}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

   x                                 x            
sin (4)*(pi*I + log(-sin(4)))   8*sin (4)*sin(8*x)
----------------------------- + ------------------
           cos(8*x)                    2          
                                    cos (8*x)

$$\frac{8 \sin^{x}{\left(4 \right)} \sin{\left(8 x \right)}}{\cos^{2}{\left(8 x \right)}} + \frac{\left(\log{\left(- \sin{\left(4 \right)} \right)} + i \pi\right) \sin^{x}{\left(4 \right)}}{\cos{\left(8 x \right)}}$$

Simplify

The second derivative [src]

        /                                     2                                         \
   x    |                          2   128*sin (8*x)   16*(pi*I + log(-sin(4)))*sin(8*x)|
sin (4)*|64 + (pi*I + log(-sin(4)))  + ------------- + ---------------------------------|
        |                                   2                       cos(8*x)            |
        \                                cos (8*x)                                      /
-----------------------------------------------------------------------------------------
                                         cos(8*x)

$$\frac{\left(\frac{128 \sin^{2}{\left(8 x \right)}}{\cos^{2}{\left(8 x \right)}} + \frac{16 \left(\log{\left(- \sin{\left(4 \right)} \right)} + i \pi\right) \sin{\left(8 x \right)}}{\cos{\left(8 x \right)}} + 64 + \left(\log{\left(- \sin{\left(4 \right)} \right)} + i \pi\right)^{2}\right) \sin^{x}{\left(4 \right)}}{\cos{\left(8 x \right)}}$$

Simplify

The third derivative [src]

        /                                                                                                                /         2     \         \
        |                                                                                                                |    6*sin (8*x)|         |
        |                                                                                                            512*|5 + -----------|*sin(8*x)|
        |                             /         2     \                                                 2                |        2      |         |
   x    |                     3       |    2*sin (8*x)|                         24*(pi*I + log(-sin(4))) *sin(8*x)       \     cos (8*x) /         |
sin (4)*|(pi*I + log(-sin(4)))  + 192*|1 + -----------|*(pi*I + log(-sin(4))) + ---------------------------------- + ------------------------------|
        |                             |        2      |                                      cos(8*x)                           cos(8*x)           |
        \                             \     cos (8*x) /                                                                                            /
----------------------------------------------------------------------------------------------------------------------------------------------------
                                                                      cos(8*x)

$$\frac{\left(192 \cdot \left(\frac{2 \sin^{2}{\left(8 x \right)}}{\cos^{2}{\left(8 x \right)}} + 1\right) \left(\log{\left(- \sin{\left(4 \right)} \right)} + i \pi\right) + \frac{512 \cdot \left(\frac{6 \sin^{2}{\left(8 x \right)}}{\cos^{2}{\left(8 x \right)}} + 5\right) \sin{\left(8 x \right)}}{\cos{\left(8 x \right)}} + \frac{24 \left(\log{\left(- \sin{\left(4 \right)} \right)} + i \pi\right)^{2} \sin{\left(8 x \right)}}{\cos{\left(8 x \right)}} + \left(\log{\left(- \sin{\left(4 \right)} \right)} + i \pi\right)^{3}\right) \sin^{x}{\left(4 \right)}}{\cos{\left(8 x \right)}}$$

Simplify

The graph

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Identical expressions

Derivative of y=sin4^x/cos(8x)

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The solution