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Derivative of:
Derivative of e^x^2
Derivative of tanh(x)
Derivative of cos²x
Derivative of (π-2x)³
Identical expressions
ctg^ four (five ^x)
ctg to the power of 4(5 to the power of x)
ctg to the power of four (five to the power of x)
ctg4(5x)
ctg45x
ctg⁴(5^x)
ctg^45^x

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

Derivative of ctg^4(5^x)

The solution

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   4/ x\
cot \5 /

$$\cot^{4}{\left(5^{x} \right)}$$

cot(5^x)^4

Detail solution

Let $u = \cot{\left(5^{x} \right)}$ .
Apply the power rule: $u^{4}$ goes to $4 u^{3}$
Then, apply the chain rule. Multiply by $\frac{d}{d x} \cot{\left(5^{x} \right)}$ :
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 $\frac{d}{d x} \tan{\left(5^{x} \right)}$ :
    1. Rewrite the function to be differentiated:
      
      $\tan{\left(5^{x} \right)} = \frac{\sin{\left(5^{x} \right)}}{\cos{\left(5^{x} \right)}}$
    2. Apply the quotient rule, which is:
      
      $\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$
      
      $f{\left(x \right)} = \sin{\left(5^{x} \right)}$ 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 $\frac{d}{d x} 5^{x}$ :
      
      $\frac{d}{d x} 5^{x} = 5^{x} \log{\left(5 \right)}$
      
      The result of the chain rule is:
      
      $5^{x} \log{\left(5 \right)} \cos{\left(5^{x} \right)}$
      
      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 $\frac{d}{d x} 5^{x}$ :
      
      $\frac{d}{d x} 5^{x} = 5^{x} \log{\left(5 \right)}$
      
      The result of the chain rule is:
      
      $- 5^{x} \log{\left(5 \right)} \sin{\left(5^{x} \right)}$
      
      Now plug in to the quotient rule:
      
      $\frac{5^{x} \log{\left(5 \right)} \sin^{2}{\left(5^{x} \right)} + 5^{x} \log{\left(5 \right)} \cos^{2}{\left(5^{x} \right)}}{\cos^{2}{\left(5^{x} \right)}}$
    The result of the chain rule is:
    
    $- \frac{5^{x} \log{\left(5 \right)} \sin^{2}{\left(5^{x} \right)} + 5^{x} \log{\left(5 \right)} \cos^{2}{\left(5^{x} \right)}}{\cos^{2}{\left(5^{x} \right)} \tan^{2}{\left(5^{x} \right)}}$
  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 $\frac{d}{d x} f{\left(x \right)}$ :
    1. Let $u = 5^{x}$ .
    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 $\frac{d}{d x} 5^{x}$ :
      
      $\frac{d}{d x} 5^{x} = 5^{x} \log{\left(5 \right)}$
      
      The result of the chain rule is:
      
      $- 5^{x} \log{\left(5 \right)} \sin{\left(5^{x} \right)}$
    To find $\frac{d}{d x} g{\left(x \right)}$ :
    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 $\frac{d}{d x} 5^{x}$ :
      
      $\frac{d}{d x} 5^{x} = 5^{x} \log{\left(5 \right)}$
      
      The result of the chain rule is:
      
      $5^{x} \log{\left(5 \right)} \cos{\left(5^{x} \right)}$
    Now plug in to the quotient rule:
    
    $\frac{- 5^{x} \log{\left(5 \right)} \sin^{2}{\left(5^{x} \right)} - 5^{x} \log{\left(5 \right)} \cos^{2}{\left(5^{x} \right)}}{\sin^{2}{\left(5^{x} \right)}}$
The result of the chain rule is:

$- \frac{4 \left(5^{x} \log{\left(5 \right)} \sin^{2}{\left(5^{x} \right)} + 5^{x} \log{\left(5 \right)} \cos^{2}{\left(5^{x} \right)}\right) \cot^{3}{\left(5^{x} \right)}}{\cos^{2}{\left(5^{x} \right)} \tan^{2}{\left(5^{x} \right)}}$
Now simplify:

$- \frac{4 \cdot 5^{x} \log{\left(5 \right)} \cos^{3}{\left(5^{x} \right)}}{\sin^{5}{\left(5^{x} \right)}}$

The answer is:

$- \frac{4 \cdot 5^{x} \log{\left(5 \right)} \cos^{3}{\left(5^{x} \right)}}{\sin^{5}{\left(5^{x} \right)}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

   x    3/ x\ /        2/ x\\       
4*5 *cot \5 /*\-1 - cot \5 //*log(5)

$$4 \cdot 5^{x} \left(- \cot^{2}{\left(5^{x} \right)} - 1\right) \log{\left(5 \right)} \cot^{3}{\left(5^{x} \right)}$$

Simplify

The second derivative [src]

   x    2/ x\    2    /       2/ x\\ /     / x\      x    2/ x\      x /       2/ x\\\
4*5 *cot \5 /*log (5)*\1 + cot \5 //*\- cot\5 / + 2*5 *cot \5 / + 3*5 *\1 + cot \5 ///

$$4 \cdot 5^{x} \left(\cot^{2}{\left(5^{x} \right)} + 1\right) \left(3 \cdot 5^{x} \left(\cot^{2}{\left(5^{x} \right)} + 1\right) + 2 \cdot 5^{x} \cot^{2}{\left(5^{x} \right)} - \cot{\left(5^{x} \right)}\right) \log{\left(5 \right)}^{2} \cot^{2}{\left(5^{x} \right)}$$

Simplify

The third derivative [src]

                            /                                  2                                                                                                  \        
   x    3    /       2/ x\\ |     2/ x\      2*x /       2/ x\\       2*x    4/ x\      x    3/ x\       2*x    2/ x\ /       2/ x\\      x /       2/ x\\    / x\|    / x\
4*5 *log (5)*\1 + cot \5 //*\- cot \5 / - 6*5   *\1 + cot \5 //  - 4*5   *cot \5 / + 6*5 *cot \5 / - 20*5   *cot \5 /*\1 + cot \5 // + 9*5 *\1 + cot \5 //*cot\5 //*cot\5 /

$$4 \cdot 5^{x} \left(\cot^{2}{\left(5^{x} \right)} + 1\right) \left(- 6 \cdot 5^{2 x} \left(\cot^{2}{\left(5^{x} \right)} + 1\right)^{2} - 20 \cdot 5^{2 x} \left(\cot^{2}{\left(5^{x} \right)} + 1\right) \cot^{2}{\left(5^{x} \right)} - 4 \cdot 5^{2 x} \cot^{4}{\left(5^{x} \right)} + 9 \cdot 5^{x} \left(\cot^{2}{\left(5^{x} \right)} + 1\right) \cot{\left(5^{x} \right)} + 6 \cdot 5^{x} \cot^{3}{\left(5^{x} \right)} - \cot^{2}{\left(5^{x} \right)}\right) \log{\left(5 \right)}^{3} \cot{\left(5^{x} \right)}$$

Simplify

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Identical expressions

Derivative of ctg^4(5^x)

The graph:

Piecewise:

The solution

Method #1

Method #2