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Identical expressions
one /(sin(5x))^ one / four
1 divide by ( sinus of (5x)) to the power of 1 divide by 4
one divide by ( sinus of (5x)) to the power of one divide by four
1/(sin(5x))1/4
1/sin5x1/4
1/sin5x^1/4
1 divide by (sin(5x))^1 divide by 4

The derivative of the function/ 1/(sin(5x))^1/4

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

The solution

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       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) /

$$\frac{d}{d x} 1 \cdot \frac{1}{\sqrt[4]{\sin{\left(5 x \right)}}}$$

Transform

Detail solution

Apply the quotient rule, which is:

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

$f{\left(x \right)} = 1$ and $g{\left(x \right)} = \sqrt[4]{\sin{\left(5 x \right)}}$ .

To find $\frac{d}{d x} f{\left(x \right)}$ :
1. The derivative of the constant $1$ is zero.
To find $\frac{d}{d x} g{\left(x \right)}$ :
1. Let $u = \sin{\left(5 x \right)}$ .
2. Apply the power rule: $\sqrt[4]{u}$ goes to $\frac{1}{4 u^{\frac{3}{4}}}$
3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \sin{\left(5 x \right)}$ :
  1. Let $u = 5 x$ .
  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 $\frac{d}{d x} 5 x$ :
    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:
    
    $5 \cos{\left(5 x \right)}$
  The result of the chain rule is:
  
  $\frac{5 \cos{\left(5 x \right)}}{4 \sin^{\frac{3}{4}}{\left(5 x \right)}}$
Now plug in to the quotient rule:

$- \frac{5 \cos{\left(5 x \right)}}{4 \sin^{\frac{5}{4}}{\left(5 x \right)}}$

The answer is:

$- \frac{5 \cos{\left(5 x \right)}}{4 \sin^{\frac{5}{4}}{\left(5 x \right)}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

      -5*cos(5*x)      
-----------------------
           4 __________
4*sin(5*x)*\/ sin(5*x)

$$- \frac{5 \cos{\left(5 x \right)}}{4 \sqrt[4]{\sin{\left(5 x \right)}} \sin{\left(5 x \right)}}$$

Simplify

The second derivative [src]

   /         2     \
   |    5*cos (5*x)|
25*|4 + -----------|
   |        2      |
   \     sin (5*x) /
--------------------
     4 __________   
  16*\/ sin(5*x)

$$\frac{25 \cdot \left(4 + \frac{5 \cos^{2}{\left(5 x \right)}}{\sin^{2}{\left(5 x \right)}}\right)}{16 \sqrt[4]{\sin{\left(5 x \right)}}}$$

Simplify

The third derivative [src]

     /           2     \         
     |     45*cos (5*x)|         
-125*|44 + ------------|*cos(5*x)
     |         2       |         
     \      sin (5*x)  /         
---------------------------------
                5/4              
          64*sin   (5*x)

$$- \frac{125 \cdot \left(44 + \frac{45 \cos^{2}{\left(5 x \right)}}{\sin^{2}{\left(5 x \right)}}\right) \cos{\left(5 x \right)}}{64 \sin^{\frac{5}{4}}{\left(5 x \right)}}$$

Simplify

The graph

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

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The solution