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Derivative of 3^(1/x)
Identical expressions
sqrt(sin^ four ((x- three)/x), four)
square root of ( sinus of to the power of 4((x minus 3) divide by x),4)
square root of ( sinus of to the power of four ((x minus three) divide by x), four)
√(sin^4((x-3)/x),4)
sqrt(sin4((x-3)/x),4)
sqrtsin4x-3/x,4
sqrt(sin⁴((x-3)/x),4)
sqrtsin^4x-3/x,4
sqrt(sin^4((x-3) divide by x),4)
Similar expressions
sqrt(sin^4((x+3)/x),4)

The derivative of the function/ sqrt(sin^4((x-3)/x),4)

Derivative of sqrt(sin^4((x-3)/x),4)

The solution

You have entered [src]

    _____________
   /    4/x - 3\ 
  /  sin |-----| 
\/       \  x  /

$$\sqrt{\frac{1}{\frac{1}{\sin^{4}{\left(\frac{x - 3}{x} \right)}}}}$$

sqrt(sin((x - 3)/x)^4)

Detail solution

Let $u = \sin^{4}{\left(\frac{x + -3}{x} \right)}$ .
Apply the power rule: $\sqrt{u}$ goes to $\frac{1}{2 \sqrt{u}}$
Then, apply the chain rule. Multiply by $\frac{d}{d x} \sin^{4}{\left(\frac{x + -3}{x} \right)}$ :
1. Let $u = \sin{\left(\frac{x + -3}{x} \right)}$ .
2. Apply the power rule: $u^{4}$ goes to $4 u^{3}$
3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \sin{\left(\frac{x + -3}{x} \right)}$ :
  1. Let $u = \frac{x + -3}{x}$ .
  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 $\frac{d}{d x} \frac{x + -3}{x}$ :
    1. Apply the quotient rule, which is:
      
      $\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$
      
      $f{\left(x \right)} = x - 3$ and $g{\left(x \right)} = x$ .
      
      To find $\frac{d}{d x} f{\left(x \right)}$ :
      1. Differentiate $x - 3$ term by term:
        
        The derivative of the constant $-3$ is zero.
        
        Apply the power rule: $x$ goes to $1$
        
        The result is: $1$
      To find $\frac{d}{d x} g{\left(x \right)}$ :
      1. Apply the power rule: $x$ goes to $1$
      Now plug in to the quotient rule:
      
      $\frac{3}{x^{2}}$
    The result of the chain rule is:
    
    $\frac{3 \cos{\left(\frac{x + -3}{x} \right)}}{x^{2}}$
  The result of the chain rule is:
  
  $\frac{12 \sin^{3}{\left(\frac{x + -3}{x} \right)} \cos{\left(\frac{x + -3}{x} \right)}}{x^{2}}$
The result of the chain rule is:

$\frac{6 \sin^{3}{\left(\frac{x + -3}{x} \right)} \cos{\left(\frac{x + -3}{x} \right)}}{x^{2} \sqrt{\sin^{4}{\left(\frac{x + -3}{x} \right)}}}$
Now simplify:

$\frac{6 \sin^{3}{\left(\frac{x - 3}{x} \right)} \cos{\left(\frac{x - 3}{x} \right)}}{x^{2} \sqrt{\sin^{4}{\left(\frac{x - 3}{x} \right)}}}$

The answer is:

$\frac{6 \sin^{3}{\left(\frac{x - 3}{x} \right)} \cos{\left(\frac{x - 3}{x} \right)}}{x^{2} \sqrt{\sin^{4}{\left(\frac{x - 3}{x} \right)}}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

      _____________                       
     /    4/x - 3\  /1   x - 3\    /x - 3\
2*  /  sin |-----| *|- - -----|*cos|-----|
  \/       \  x  /  |x      2 |    \  x  /
                    \      x  /           
------------------------------------------
                   /x - 3\                
                sin|-----|                
                   \  x  /

$$\frac{2 \left(\frac{1}{x} - \frac{x + -3}{x^{2}}\right) \sqrt{\sin^{4}{\left(\frac{x + -3}{x} \right)}} \cos{\left(\frac{x + -3}{x} \right)}}{\sin{\left(\frac{x + -3}{x} \right)}}$$

Simplify

The second derivative [src]

                                  /                   /-3 + x\      2/-3 + x\ /    -3 + x\\
      ______________              |              2*cos|------|   cos |------|*|1 - ------||
     /    4/-3 + x\  /    -3 + x\ |     -3 + x        \  x   /       \  x   / \      x   /|
2*  /  sin |------| *|1 - ------|*|-1 + ------ - ------------- + -------------------------|
  \/       \  x   /  \      x   / |       x          /-3 + x\              2/-3 + x\      |
                                  |               sin|------|           sin |------|      |
                                  \                  \  x   /               \  x   /      /
-------------------------------------------------------------------------------------------
                                              2                                            
                                             x

$$\frac{2 \left(1 - \frac{x - 3}{x}\right) \left(\frac{\left(1 - \frac{x - 3}{x}\right) \cos^{2}{\left(\frac{x - 3}{x} \right)}}{\sin^{2}{\left(\frac{x - 3}{x} \right)}} - 1 - \frac{2 \cos{\left(\frac{x - 3}{x} \right)}}{\sin{\left(\frac{x - 3}{x} \right)}} + \frac{x - 3}{x}\right) \sqrt{\sin^{4}{\left(\frac{x - 3}{x} \right)}}}{x^{2}}$$

Simplify

The third derivative [src]

                                  /                                                                             2            \
                                  |                      /-3 + x\        2/-3 + x\ /    -3 + x\     /    -3 + x\     /-3 + x\|
      ______________              |                 3*cos|------|   3*cos |------|*|1 - ------|   2*|1 - ------| *cos|------||
     /    4/-3 + x\  /    -3 + x\ |    3*(-3 + x)        \  x   /         \  x   / \      x   /     \      x   /     \  x   /|
4*  /  sin |------| *|1 - ------|*|3 - ---------- + ------------- - --------------------------- - ---------------------------|
  \/       \  x   /  \      x   / |        x            /-3 + x\               2/-3 + x\                     /-3 + x\        |
                                  |                  sin|------|            sin |------|                  sin|------|        |
                                  \                     \  x   /                \  x   /                     \  x   /        /
------------------------------------------------------------------------------------------------------------------------------
                                                               3                                                              
                                                              x

/ 2 \ | /-3 + x\ 2/-3 + x\ / -3 + x\ / -3 + x\ /-3 + x\| ______________ | 3*cos|------| 3*cos |------|*|1 - ------| 2*|1 - ------| *cos|------|| / 4/-3 + x\ / -3 + x\ | 3*(-3 + x) \ x / \ x / \ x / \ x / \ x /| 4* / sin |------| *|1 - ------|*|3 - ---------- + ------------- - --------------------------- - ---------------------------| \/ \ x / \ x / | x /-3 + x\ 2/-3 + x\ /-3 + x\ | | sin|------| sin |------| sin|------| | \ \ x / \ x / \ x / / ------------------------------------------------------------------------------------------------------------------------------ 3 x

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Derivative of sqrt(sin^4((x-3)/x),4)

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