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(sin3x)/cos(3x)^5

Derivative of (sin3x)/cos(3x)^5

Function f() - derivative -N order at the point
v

The graph:

from to

Piecewise:

The solution

You have entered [src]
 sin(3*x)
---------
   5     
cos (3*x)
$$\frac{\sin{\left(3 x \right)}}{\cos^{5}{\left(3 x \right)}}$$
d / sin(3*x)\
--|---------|
dx|   5     |
  \cos (3*x)/
$$\frac{d}{d x} \frac{\sin{\left(3 x \right)}}{\cos^{5}{\left(3 x \right)}}$$
Detail solution
  1. Apply the quotient rule, which is:

    and .

    To find :

    1. Let .

    2. The derivative of sine is cosine:

    3. Then, apply the chain rule. Multiply by :

      1. The derivative of a constant times a function is the constant times the derivative of the function.

        1. Apply the power rule: goes to

        So, the result is:

      The result of the chain rule is:

    To find :

    1. Let .

    2. Apply the power rule: goes to

    3. Then, apply the chain rule. Multiply by :

      1. Let .

      2. The derivative of cosine is negative sine:

      3. Then, apply the chain rule. Multiply by :

        1. The derivative of a constant times a function is the constant times the derivative of the function.

          1. Apply the power rule: goes to

          So, the result is:

        The result of the chain rule is:

      The result of the chain rule is:

    Now plug in to the quotient rule:

  2. Now simplify:


The answer is:

The graph
The first derivative [src]
                   2     
3*cos(3*x)   15*sin (3*x)
---------- + ------------
   5             6       
cos (3*x)     cos (3*x)  
$$\frac{15 \sin^{2}{\left(3 x \right)}}{\cos^{6}{\left(3 x \right)}} + \frac{3 \cos{\left(3 x \right)}}{\cos^{5}{\left(3 x \right)}}$$
The second derivative [src]
  /           2     \         
  |     30*sin (3*x)|         
9*|14 + ------------|*sin(3*x)
  |         2       |         
  \      cos (3*x)  /         
------------------------------
             5                
          cos (3*x)           
$$\frac{9 \cdot \left(\frac{30 \sin^{2}{\left(3 x \right)}}{\cos^{2}{\left(3 x \right)}} + 14\right) \sin{\left(3 x \right)}}{\cos^{5}{\left(3 x \right)}}$$
The third derivative [src]
   /                                /           2     \\
   |                         2      |     42*sin (3*x)||
   |                    5*sin (3*x)*|17 + ------------||
   |           2                    |         2       ||
   |     75*sin (3*x)               \      cos (3*x)  /|
27*|14 + ------------ + -------------------------------|
   |         2                        2                |
   \      cos (3*x)                cos (3*x)           /
--------------------------------------------------------
                          4                             
                       cos (3*x)                        
$$\frac{27 \cdot \left(\frac{5 \cdot \left(\frac{42 \sin^{2}{\left(3 x \right)}}{\cos^{2}{\left(3 x \right)}} + 17\right) \sin^{2}{\left(3 x \right)}}{\cos^{2}{\left(3 x \right)}} + \frac{75 \sin^{2}{\left(3 x \right)}}{\cos^{2}{\left(3 x \right)}} + 14\right)}{\cos^{4}{\left(3 x \right)}}$$
The graph
Derivative of (sin3x)/cos(3x)^5