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Identical expressions
cot(three *x)^(two)
cotangent of (3 multiply by x) to the power of (2)
cotangent of (three multiply by x) to the power of (two)
cot(3*x)(2)
cot3*x2
cot(3x)^(2)
cot(3x)(2)
cot3x2
cot3x^2

The derivative of the function/ cot(3*x)^(2)

Derivative of cot(3*x)^(2)

The solution

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   2     
cot (3*x)

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

cot(3*x)^2

Detail solution

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

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

/          2     \         
\-6 - 6*cot (3*x)/*cot(3*x)

$$\left(- 6 \cot^{2}{\left(3 x \right)} - 6\right) \cot{\left(3 x \right)}$$

Simplify

The second derivative [src]

   /       2     \ /         2     \
18*\1 + cot (3*x)/*\1 + 3*cot (3*x)/

$$18 \left(\cot^{2}{\left(3 x \right)} + 1\right) \left(3 \cot^{2}{\left(3 x \right)} + 1\right)$$

Simplify

The third derivative [src]

     /       2     \ /         2     \         
-216*\1 + cot (3*x)/*\2 + 3*cot (3*x)/*cot(3*x)

$$- 216 \left(\cot^{2}{\left(3 x \right)} + 1\right) \left(3 \cot^{2}{\left(3 x \right)} + 2\right) \cot{\left(3 x \right)}$$

Simplify

The graph

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Identical expressions

Derivative of cot(3*x)^(2)

The graph:

Piecewise:

The solution

Method #1

Method #2