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Derivative of:
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Derivative of -e^x
Derivative of e^(2-x)
Derivative of x^2-4*x
Identical expressions
e^ctg5x/(three x-4x+ two)^3
e to the power of ctg5x divide by (3x minus 4x plus 2) cubed
e to the power of ctg5x divide by (three x minus 4x plus two) cubed
ectg5x/(3x-4x+2)3
ectg5x/3x-4x+23
e^ctg5x/(3x-4x+2)³
e to the power of ctg5x/(3x-4x+2) to the power of 3
e^ctg5x/3x-4x+2^3
e^ctg5x divide by (3x-4x+2)^3
Similar expressions
e^ctg5x/(3x+4x+2)^3
e^ctg5x/(3x-4x-2)^3

The derivative of the function/ e^ctg5x/(3x-4x+2)^3

Derivative of e^ctg5x/(3x-4x+2)^3

The solution

You have entered [src]

    cot(5*x)    
   e            
----------------
               3
(3*x - 4*x + 2)

$$\frac{e^{\cot{\left(5 x \right)}}}{\left(- 4 x + 3 x + 2\right)^{3}}$$

  /    cot(5*x)    \
d |   e            |
--|----------------|
dx|               3|
  \(3*x - 4*x + 2) /

$$\frac{d}{d x} \frac{e^{\cot{\left(5 x \right)}}}{\left(- 4 x + 3 x + 2\right)^{3}}$$

Transform

Detail solution

Apply the quotient rule, which is:

$\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$

$f{\left(x \right)} = e^{\cot{\left(5 x \right)}}$ and $g{\left(x \right)} = \left(2 - x\right)^{3}$ .

To find $\frac{d}{d x} f{\left(x \right)}$ :
1. Let $u = \cot{\left(5 x \right)}$ .
2. The derivative of $e^{u}$ is itself.
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 $\frac{d}{d x} \tan{\left(5 x \right)}$ :
      
      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 of the chain rule is:
  
  $- \frac{\left(5 \sin^{2}{\left(5 x \right)} + 5 \cos^{2}{\left(5 x \right)}\right) e^{\cot{\left(5 x \right)}}}{\cos^{2}{\left(5 x \right)} \tan^{2}{\left(5 x \right)}}$
To find $\frac{d}{d x} g{\left(x \right)}$ :
1. Let $u = 2 - x$ .
2. Apply the power rule: $u^{3}$ goes to $3 u^{2}$
3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(2 - x\right)$ :
  1. Differentiate $2 - x$ term by term:
    1. The derivative of the constant $2$ is zero.
    2. The derivative of a constant times a function is the constant times the derivative of the function.
      1. Apply the power rule: $x$ goes to $1$
      So, the result is: $-1$
    The result is: $-1$
  The result of the chain rule is:
  
  $- 3 \left(2 - x\right)^{2}$
Now plug in to the quotient rule:

$\frac{- \frac{\left(2 - x\right)^{3} \cdot \left(5 \sin^{2}{\left(5 x \right)} + 5 \cos^{2}{\left(5 x \right)}\right) e^{\cot{\left(5 x \right)}}}{\cos^{2}{\left(5 x \right)} \tan^{2}{\left(5 x \right)}} + 3 \left(2 - x\right)^{2} e^{\cot{\left(5 x \right)}}}{\left(2 - x\right)^{6}}$
Now simplify:

$\frac{\left(\frac{5 x}{\sin^{2}{\left(5 x \right)}} + 3 - \frac{10}{\sin^{2}{\left(5 x \right)}}\right) e^{\frac{1}{\tan{\left(5 x \right)}}}}{\left(x - 2\right)^{4}}$

The answer is:

$\frac{\left(\frac{5 x}{\sin^{2}{\left(5 x \right)}} + 3 - \frac{10}{\sin^{2}{\left(5 x \right)}}\right) e^{\frac{1}{\tan{\left(5 x \right)}}}}{\left(x - 2\right)^{4}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

$$\frac{\left(- 5 \cot^{2}{\left(5 x \right)} - 5\right) e^{\cot{\left(5 x \right)}}}{\left(- 4 x + 3 x + 2\right)^{3}} + \frac{3 e^{\cot{\left(5 x \right)}}}{\left(- 4 x + 3 x + 2\right)^{4}}$$

Simplify

The second derivative [src]

 /                                                                 /       2     \\           
 |    12         /       2     \ /       2                  \   30*\1 + cot (5*x)/|  cot(5*x) 
-|--------- + 25*\1 + cot (5*x)/*\1 + cot (5*x) + 2*cot(5*x)/ + ------------------|*e         
 |        2                                                           -2 + x      |           
 \(-2 + x)                                                                        /           
----------------------------------------------------------------------------------------------
                                                  3                                           
                                          (-2 + x)

$$- \frac{\left(25 \left(\cot^{2}{\left(5 x \right)} + 1\right) \left(\cot^{2}{\left(5 x \right)} + 2 \cot{\left(5 x \right)} + 1\right) + \frac{30 \left(\cot^{2}{\left(5 x \right)} + 1\right)}{x - 2} + \frac{12}{\left(x - 2\right)^{2}}\right) e^{\cot{\left(5 x \right)}}}{\left(x - 2\right)^{3}}$$

Simplify

The third derivative [src]

  /                               /                   2                                           \      /       2     \      /       2     \ /       2                  \\          
  |    12         /       2     \ |    /       2     \         2          /       2     \         |   36*\1 + cot (5*x)/   45*\1 + cot (5*x)/*\1 + cot (5*x) + 2*cot(5*x)/|  cot(5*x)
5*|--------- + 25*\1 + cot (5*x)/*\2 + \1 + cot (5*x)/  + 6*cot (5*x) + 6*\1 + cot (5*x)/*cot(5*x)/ + ------------------ + -----------------------------------------------|*e        
  |        3                                                                                                      2                             -2 + x                    |          
  \(-2 + x)                                                                                               (-2 + x)                                                        /          
-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
                                                                                              3                                                                                      
                                                                                      (-2 + x)

$$\frac{5 \cdot \left(25 \left(\cot^{2}{\left(5 x \right)} + 1\right) \left(\left(\cot^{2}{\left(5 x \right)} + 1\right)^{2} + 6 \left(\cot^{2}{\left(5 x \right)} + 1\right) \cot{\left(5 x \right)} + 6 \cot^{2}{\left(5 x \right)} + 2\right) + \frac{45 \left(\cot^{2}{\left(5 x \right)} + 1\right) \left(\cot^{2}{\left(5 x \right)} + 2 \cot{\left(5 x \right)} + 1\right)}{x - 2} + \frac{36 \left(\cot^{2}{\left(5 x \right)} + 1\right)}{\left(x - 2\right)^{2}} + \frac{12}{\left(x - 2\right)^{3}}\right) e^{\cot{\left(5 x \right)}}}{\left(x - 2\right)^{3}}$$

Simplify

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Derivative of e^ctg5x/(3x-4x+2)^3

The graph:

Piecewise:

The solution

Method #1

Method #2