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Derivative of:
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Derivative of e^sin(x)
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Derivative of e^(-2*x)
Identical expressions
-(((ctgx)^ three)/ three)
minus (((ctgx) cubed ) divide by 3)
minus (((ctgx) to the power of three) divide by three)
-(((ctgx)3)/3)
-ctgx3/3
-(((ctgx)³)/3)
-(((ctgx) to the power of 3)/3)
-ctgx^3/3
-(((ctgx)^3) divide by 3)
Similar expressions
(((ctgx)^3)/3)

The derivative of the function/ -(((ctgx)^3)/3)

Derivative of -(((ctgx)^3)/3)

The solution

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    3    
-cot (x) 
---------
    3

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

  /    3    \
d |-cot (x) |
--|---------|
dx\    3    /

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

Transform

Detail solution

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

$\frac{\cos^{2}{\left(x \right)}}{\sin^{4}{\left(x \right)}}$

The answer is:

$\frac{\cos^{2}{\left(x \right)}}{\sin^{4}{\left(x \right)}}$

The graph

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The first derivative [src]

    2    /          2   \ 
-cot (x)*\-3 - 3*cot (x)/ 
--------------------------
            3

$$- \frac{\left(- 3 \cot^{2}{\left(x \right)} - 3\right) \cot^{2}{\left(x \right)}}{3}$$

Simplify

The second derivative [src]

   /       2   \ /         2   \       
-2*\1 + cot (x)/*\1 + 2*cot (x)/*cot(x)

$$- 2 \left(\cot^{2}{\left(x \right)} + 1\right) \left(2 \cot^{2}{\left(x \right)} + 1\right) \cot{\left(x \right)}$$

Simplify

The third derivative [src]

                /             2                                      \
  /       2   \ |/       2   \         4           2    /       2   \|
2*\1 + cot (x)/*\\1 + cot (x)/  + 2*cot (x) + 7*cot (x)*\1 + cot (x)//

$$2 \left(\cot^{2}{\left(x \right)} + 1\right) \left(\left(\cot^{2}{\left(x \right)} + 1\right)^{2} + 7 \left(\cot^{2}{\left(x \right)} + 1\right) \cot^{2}{\left(x \right)} + 2 \cot^{4}{\left(x \right)}\right)$$

Simplify

The graph

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Identical expressions

Similar expressions

Derivative of -(((ctgx)^3)/3)

The graph:

Piecewise:

The solution

Method #1

Method #2