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1/(sin(5x))^1/4

Derivative of 1/(sin(5x))^1/4

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

The graph:

from to

Piecewise:

The solution

You have entered [src]
       1      
1*------------
  4 __________
  \/ sin(5*x) 
$$1 \cdot \frac{1}{\sqrt[4]{\sin{\left(5 x \right)}}}$$
d /       1      \
--|1*------------|
dx|  4 __________|
  \  \/ sin(5*x) /
$$\frac{d}{d x} 1 \cdot \frac{1}{\sqrt[4]{\sin{\left(5 x \right)}}}$$
Detail solution
  1. Apply the quotient rule, which is:

    and .

    To find :

    1. The derivative of the constant is zero.

    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 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:

      The result of the chain rule is:

    Now plug in to the quotient rule:


The answer is:

The graph
The first derivative [src]
      -5*cos(5*x)      
-----------------------
           4 __________
4*sin(5*x)*\/ sin(5*x) 
$$- \frac{5 \cos{\left(5 x \right)}}{4 \sqrt[4]{\sin{\left(5 x \right)}} \sin{\left(5 x \right)}}$$
The second derivative [src]
   /         2     \
   |    5*cos (5*x)|
25*|4 + -----------|
   |        2      |
   \     sin (5*x) /
--------------------
     4 __________   
  16*\/ sin(5*x)    
$$\frac{25 \cdot \left(4 + \frac{5 \cos^{2}{\left(5 x \right)}}{\sin^{2}{\left(5 x \right)}}\right)}{16 \sqrt[4]{\sin{\left(5 x \right)}}}$$
The third derivative [src]
     /           2     \         
     |     45*cos (5*x)|         
-125*|44 + ------------|*cos(5*x)
     |         2       |         
     \      sin (5*x)  /         
---------------------------------
                5/4              
          64*sin   (5*x)         
$$- \frac{125 \cdot \left(44 + \frac{45 \cos^{2}{\left(5 x \right)}}{\sin^{2}{\left(5 x \right)}}\right) \cos{\left(5 x \right)}}{64 \sin^{\frac{5}{4}}{\left(5 x \right)}}$$
The graph
Derivative of 1/(sin(5x))^1/4