Derivative of sin8(x)/cos(5x)

The solution

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   8    
sin (x) 
--------
cos(5*x)

$$\frac{\sin^{8}{\left(x \right)}}{\cos{\left(5 x \right)}}$$

sin(x)^8/cos(5*x)

Detail solution

Apply the quotient rule, which is:

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

$f{\left(x \right)} = \sin^{8}{\left(x \right)}$ and $g{\left(x \right)} = \cos{\left(5 x \right)}$ .

To find $\frac{d}{d x} f{\left(x \right)}$ :
1. Let $u = \sin{\left(x \right)}$ .
2. Apply the power rule: $u^{8}$ goes to $8 u^{7}$
3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \sin{\left(x \right)}$ :
  1. The derivative of sine is cosine:
    
    $\frac{d}{d x} \sin{\left(x \right)} = \cos{\left(x \right)}$
  The result of the chain rule is:
  
  $8 \sin^{7}{\left(x \right)} \cos{\left(x \right)}$
To find $\frac{d}{d x} g{\left(x \right)}$ :
1. Let $u = 5 x$ .
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 $\frac{d}{d x} 5 x$ :
  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: $5$
  The result of the chain rule is:
  
  $- 5 \sin{\left(5 x \right)}$
Now plug in to the quotient rule:

$\frac{5 \sin^{8}{\left(x \right)} \sin{\left(5 x \right)} + 8 \sin^{7}{\left(x \right)} \cos{\left(x \right)} \cos{\left(5 x \right)}}{\cos^{2}{\left(5 x \right)}}$
Now simplify:

$\frac{\left(13 \cos{\left(4 x \right)} + 3 \cos{\left(6 x \right)}\right) \sin^{7}{\left(x \right)}}{2 \cos^{2}{\left(5 x \right)}}$

The answer is:

$\frac{\left(13 \cos{\left(4 x \right)} + 3 \cos{\left(6 x \right)}\right) \sin^{7}{\left(x \right)}}{2 \cos^{2}{\left(5 x \right)}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

     8                    7          
5*sin (x)*sin(5*x)   8*sin (x)*cos(x)
------------------ + ----------------
       2                 cos(5*x)    
    cos (5*x)

$$\frac{5 \sin^{8}{\left(x \right)} \sin{\left(5 x \right)}}{\cos^{2}{\left(5 x \right)}} + \frac{8 \sin^{7}{\left(x \right)} \cos{\left(x \right)}}{\cos{\left(5 x \right)}}$$

Simplify

The second derivative [src]

        /                                      /         2     \                            \
   6    |       2            2            2    |    2*sin (5*x)|   80*cos(x)*sin(x)*sin(5*x)|
sin (x)*|- 8*sin (x) + 56*cos (x) + 25*sin (x)*|1 + -----------| + -------------------------|
        |                                      |        2      |            cos(5*x)        |
        \                                      \     cos (5*x) /                            /
---------------------------------------------------------------------------------------------
                                           cos(5*x)

$$\frac{\left(25 \left(\frac{2 \sin^{2}{\left(5 x \right)}}{\cos^{2}{\left(5 x \right)}} + 1\right) \sin^{2}{\left(x \right)} - 8 \sin^{2}{\left(x \right)} + \frac{80 \sin{\left(x \right)} \sin{\left(5 x \right)} \cos{\left(x \right)}}{\cos{\left(5 x \right)}} + 56 \cos^{2}{\left(x \right)}\right) \sin^{6}{\left(x \right)}}{\cos{\left(5 x \right)}}$$

Simplify

The third derivative [src]

        /                                                                                                                                         /         2     \         \
        |                                                                                                                                    3    |    6*sin (5*x)|         |
        |                                                                                                                             125*sin (x)*|5 + -----------|*sin(5*x)|
        |                                                      /         2     \              /   2           2   \                               |        2      |         |
   5    |     /        2            2   \                 2    |    2*sin (5*x)|          120*\sin (x) - 7*cos (x)/*sin(x)*sin(5*x)               \     cos (5*x) /         |
sin (x)*|- 16*\- 21*cos (x) + 11*sin (x)/*cos(x) + 600*sin (x)*|1 + -----------|*cos(x) - ----------------------------------------- + --------------------------------------|
        |                                                      |        2      |                           cos(5*x)                                  cos(5*x)               |
        \                                                      \     cos (5*x) /                                                                                            /
-----------------------------------------------------------------------------------------------------------------------------------------------------------------------------
                                                                                   cos(5*x)

$$\frac{\left(600 \left(\frac{2 \sin^{2}{\left(5 x \right)}}{\cos^{2}{\left(5 x \right)}} + 1\right) \sin^{2}{\left(x \right)} \cos{\left(x \right)} + \frac{125 \left(\frac{6 \sin^{2}{\left(5 x \right)}}{\cos^{2}{\left(5 x \right)}} + 5\right) \sin^{3}{\left(x \right)} \sin{\left(5 x \right)}}{\cos{\left(5 x \right)}} - \frac{120 \left(\sin^{2}{\left(x \right)} - 7 \cos^{2}{\left(x \right)}\right) \sin{\left(x \right)} \sin{\left(5 x \right)}}{\cos{\left(5 x \right)}} - 16 \left(11 \sin^{2}{\left(x \right)} - 21 \cos^{2}{\left(x \right)}\right) \cos{\left(x \right)}\right) \sin^{5}{\left(x \right)}}{\cos{\left(5 x \right)}}$$

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Derivative of sin8(x)/cos(5x)

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