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Identical expressions
y=x^(three / eight)*ctg five x-4x^(-5)
y equally x to the power of (3 divide by 8) multiply by ctg5x minus 4x to the power of ( minus 5)
y equally x to the power of (three divide by eight) multiply by ctg five x minus 4x to the power of ( minus 5)
y=x(3/8)*ctg5x-4x(-5)
y=x3/8*ctg5x-4x-5
y=x^(3/8)ctg5x-4x^(-5)
y=x(3/8)ctg5x-4x(-5)
y=x3/8ctg5x-4x-5
y=x^3/8ctg5x-4x^-5
y=x^(3 divide by 8)*ctg5x-4x^(-5)
Similar expressions
y=x^(3/8)*ctg5x-4x^(5)
y=x^(3/8)*ctg5x+4x^(-5)

The derivative of the function/ y=x^(3/8)*ctg5x-4x^(-5)

Derivative of y=x^(3/8)*ctg5x-4x^(-5)

The solution

You have entered [src]

 3/8            4 
x   *cot(5*x) - --
                 5
                x

$$x^{\frac{3}{8}} \cot{\left(5 x \right)} - \frac{4}{x^{5}}$$

d / 3/8            4 \
--|x   *cot(5*x) - --|
dx|                 5|
  \                x /

$$\frac{d}{d x} \left(x^{\frac{3}{8}} \cot{\left(5 x \right)} - \frac{4}{x^{5}}\right)$$

Transform

Detail solution

Differentiate $x^{\frac{3}{8}} \cot{\left(5 x \right)} - \frac{4}{x^{5}}$ term by term:
1. Apply the product rule:
  
  $\frac{d}{d x} f{\left(x \right)} g{\left(x \right)} = f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}$
  
  $f{\left(x \right)} = x^{\frac{3}{8}}$ ; to find $\frac{d}{d x} f{\left(x \right)}$ :
  1. Apply the power rule: $x^{\frac{3}{8}}$ goes to $\frac{3}{8 x^{\frac{5}{8}}}$
  $g{\left(x \right)} = \cot{\left(5 x \right)}$ ; to find $\frac{d}{d x} g{\left(x \right)}$ :
  1. There are multiple ways to do this derivative.
    Method #1
    1. Rewrite the function to be differentiated:
      
      $\cot{\left(5 x \right)} = \frac{1}{\tan{\left(5 x \right)}}$
    2. Let $u = \tan{\left(5 x \right)}$ .
    3. Apply the power rule: $\frac{1}{u}$ goes to $- \frac{1}{u^{2}}$
    4. Then, apply the chain rule. Multiply by $\frac{d}{d x} \tan{\left(5 x \right)}$ :
      
      Rewrite the function to be differentiated:
      
      $\tan{\left(5 x \right)} = \frac{\sin{\left(5 x \right)}}{\cos{\left(5 x \right)}}$
      
      Apply the quotient rule, which is:
      
      $\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$
      
      $f{\left(x \right)} = \sin{\left(5 x \right)}$ and $g{\left(x \right)} = \cos{\left(5 x \right)}$ .
      
      To find $\frac{d}{d x} f{\left(x \right)}$ :
      
      Let $u = 5 x$ .
      
      The derivative of sine is cosine:
      
      $\frac{d}{d u} \sin{\left(u \right)} = \cos{\left(u \right)}$
      
      Then, apply the chain rule. Multiply by $\frac{d}{d x} 5 x$ :
      
      The derivative of a constant times a function is the constant times the derivative of the function.
      
      Apply the power rule: $x$ goes to $1$
      
      So, the result is: $5$
      
      The result of the chain rule is:
      
      $5 \cos{\left(5 x \right)}$
      
      To find $\frac{d}{d x} g{\left(x \right)}$ :
      
      Let $u = 5 x$ .
      
      The derivative of cosine is negative sine:
      
      $\frac{d}{d u} \cos{\left(u \right)} = - \sin{\left(u \right)}$
      
      Then, apply the chain rule. Multiply by $\frac{d}{d x} 5 x$ :
      
      The derivative of a constant times a function is the constant times the derivative of the function.
      
      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:
      
      $\frac{5 \sin^{2}{\left(5 x \right)} + 5 \cos^{2}{\left(5 x \right)}}{\cos^{2}{\left(5 x \right)}}$
      
      The result of the chain rule is:
      
      $- \frac{5 \sin^{2}{\left(5 x \right)} + 5 \cos^{2}{\left(5 x \right)}}{\cos^{2}{\left(5 x \right)} \tan^{2}{\left(5 x \right)}}$
    Method #2
    1. Rewrite the function to be differentiated:
      
      $\cot{\left(5 x \right)} = \frac{\cos{\left(5 x \right)}}{\sin{\left(5 x \right)}}$
    2. Apply the quotient rule, which is:
      
      $\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$
      
      $f{\left(x \right)} = \cos{\left(5 x \right)}$ and $g{\left(x \right)} = \sin{\left(5 x \right)}$ .
      
      To find $\frac{d}{d x} f{\left(x \right)}$ :
      
      Let $u = 5 x$ .
      
      The derivative of cosine is negative sine:
      
      $\frac{d}{d u} \cos{\left(u \right)} = - \sin{\left(u \right)}$
      
      Then, apply the chain rule. Multiply by $\frac{d}{d x} 5 x$ :
      
      The derivative of a constant times a function is the constant times the derivative of the function.
      
      Apply the power rule: $x$ goes to $1$
      
      So, the result is: $5$
      
      The result of the chain rule is:
      
      $- 5 \sin{\left(5 x \right)}$
      
      To find $\frac{d}{d x} g{\left(x \right)}$ :
      
      Let $u = 5 x$ .
      
      The derivative of sine is cosine:
      
      $\frac{d}{d u} \sin{\left(u \right)} = \cos{\left(u \right)}$
      
      Then, apply the chain rule. Multiply by $\frac{d}{d x} 5 x$ :
      
      The derivative of a constant times a function is the constant times the derivative of the function.
      
      Apply the power rule: $x$ goes to $1$
      
      So, the result is: $5$
      
      The result of the chain rule is:
      
      $5 \cos{\left(5 x \right)}$
      
      Now plug in to the quotient rule:
      
      $\frac{- 5 \sin^{2}{\left(5 x \right)} - 5 \cos^{2}{\left(5 x \right)}}{\sin^{2}{\left(5 x \right)}}$
  The result is: $- \frac{x^{\frac{3}{8}} \cdot \left(5 \sin^{2}{\left(5 x \right)} + 5 \cos^{2}{\left(5 x \right)}\right)}{\cos^{2}{\left(5 x \right)} \tan^{2}{\left(5 x \right)}} + \frac{3 \cot{\left(5 x \right)}}{8 x^{\frac{5}{8}}}$
2. The derivative of a constant times a function is the constant times the derivative of the function.
  1. The derivative of a constant times a function is the constant times the derivative of the function.
    1. Apply the power rule: $\frac{1}{x^{5}}$ goes to $- \frac{5}{x^{6}}$
    So, the result is: $- \frac{20}{x^{6}}$
  So, the result is: $\frac{20}{x^{6}}$
The result is: $- \frac{x^{\frac{3}{8}} \cdot \left(5 \sin^{2}{\left(5 x \right)} + 5 \cos^{2}{\left(5 x \right)}\right)}{\cos^{2}{\left(5 x \right)} \tan^{2}{\left(5 x \right)}} + \frac{20}{x^{6}} + \frac{3 \cot{\left(5 x \right)}}{8 x^{\frac{5}{8}}}$
Now simplify:

$- \frac{5 x^{\frac{3}{8}}}{\sin^{2}{\left(5 x \right)}} + \frac{20}{x^{6}} + \frac{3}{8 x^{\frac{5}{8}} \tan{\left(5 x \right)}}$

The answer is:

$- \frac{5 x^{\frac{3}{8}}}{\sin^{2}{\left(5 x \right)}} + \frac{20}{x^{6}} + \frac{3}{8 x^{\frac{5}{8}} \tan{\left(5 x \right)}}$

The graph

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Plot the graph f'(x)

The first derivative [src]

20    3/8 /          2     \   3*cot(5*x)
-- + x   *\-5 - 5*cot (5*x)/ + ----------
 6                                  5/8  
x                                8*x

$$x^{\frac{3}{8}} \left(- 5 \cot^{2}{\left(5 x \right)} - 5\right) + \frac{20}{x^{6}} + \frac{3 \cot{\left(5 x \right)}}{8 x^{\frac{5}{8}}}$$

Simplify

The second derivative [src]

  /         /       2     \                                                \
  |  24   3*\1 + cot (5*x)/   3*cot(5*x)       3/8 /       2     \         |
5*|- -- - ----------------- - ---------- + 10*x   *\1 + cot (5*x)/*cot(5*x)|
  |   7            5/8             13/8                                    |
  \  x          4*x            64*x                                        /

$$5 \cdot \left(10 x^{\frac{3}{8}} \left(\cot^{2}{\left(5 x \right)} + 1\right) \cot{\left(5 x \right)} - \frac{24}{x^{7}} - \frac{3 \left(\cot^{2}{\left(5 x \right)} + 1\right)}{4 x^{\frac{5}{8}}} - \frac{3 \cot{\left(5 x \right)}}{64 x^{\frac{13}{8}}}\right)$$

Simplify

The third derivative [src]

  /                             2                    /       2     \                                           /       2     \         \
  |168       3/8 /       2     \    39*cot(5*x)   45*\1 + cot (5*x)/        3/8    2      /       2     \   45*\1 + cot (5*x)/*cot(5*x)|
5*|--- - 50*x   *\1 + cot (5*x)/  + ----------- + ------------------ - 100*x   *cot (5*x)*\1 + cot (5*x)/ + ---------------------------|
  |  8                                    21/8             13/8                                                           5/8          |
  \ x                                512*x             64*x                                                            4*x             /

$$5 \left(- 50 x^{\frac{3}{8}} \left(\cot^{2}{\left(5 x \right)} + 1\right)^{2} - 100 x^{\frac{3}{8}} \left(\cot^{2}{\left(5 x \right)} + 1\right) \cot^{2}{\left(5 x \right)} + \frac{168}{x^{8}} + \frac{45 \left(\cot^{2}{\left(5 x \right)} + 1\right) \cot{\left(5 x \right)}}{4 x^{\frac{5}{8}}} + \frac{45 \left(\cot^{2}{\left(5 x \right)} + 1\right)}{64 x^{\frac{13}{8}}} + \frac{39 \cot{\left(5 x \right)}}{512 x^{\frac{21}{8}}}\right)$$

Simplify

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Derivative of y=x^(3/8)*ctg5x-4x^(-5)

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Piecewise:

The solution

Method #1

Method #2