Derivative of y=ln4x*e^ctgx

The solution

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          cot(x)
log(4*x)*E

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

log(4*x)*E^cot(x)

Detail solution

Apply the product rule:

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

$f{\left(x \right)} = \log{\left(4 x \right)}$ ; to find $\frac{d}{d x} f{\left(x \right)}$ :
1. Let $u = 4 x$ .
2. The derivative of $\log{\left(u \right)}$ is $\frac{1}{u}$ .
3. Then, apply the chain rule. Multiply by $\frac{d}{d x} 4 x$ :
  1. 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: $4$
  The result of the chain rule is:
  
  $\frac{1}{x}$
$g{\left(x \right)} = e^{\cot{\left(x \right)}}$ ; to find $\frac{d}{d x} g{\left(x \right)}$ :
1. Let $u = \cot{\left(x \right)}$ .
2. The derivative of $e^{u}$ is itself.
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:
  
  $- \frac{\left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right) e^{\cot{\left(x \right)}}}{\cos^{2}{\left(x \right)} \tan^{2}{\left(x \right)}}$
The result is: $- \frac{\left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right) e^{\cot{\left(x \right)}} \log{\left(4 x \right)}}{\cos^{2}{\left(x \right)} \tan^{2}{\left(x \right)}} + \frac{e^{\cot{\left(x \right)}}}{x}$
Now simplify:

$\frac{\left(- \frac{x \log{\left(x \right)}}{\sin^{2}{\left(x \right)}} - \frac{2 x \log{\left(2 \right)}}{\sin^{2}{\left(x \right)}} + 1\right) e^{\frac{1}{\tan{\left(x \right)}}}}{x}$

The answer is:

$\frac{\left(- \frac{x \log{\left(x \right)}}{\sin^{2}{\left(x \right)}} - \frac{2 x \log{\left(2 \right)}}{\sin^{2}{\left(x \right)}} + 1\right) e^{\frac{1}{\tan{\left(x \right)}}}}{x}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

$$\left(- \cot^{2}{\left(x \right)} - 1\right) e^{\cot{\left(x \right)}} \log{\left(4 x \right)} + \frac{e^{\cot{\left(x \right)}}}{x}$$

Simplify

The second derivative [src]

/         /       2   \                                                  \        
|  1    2*\1 + cot (x)/   /       2   \ /       2              \         |  cot(x)
|- -- - --------------- + \1 + cot (x)/*\1 + cot (x) + 2*cot(x)/*log(4*x)|*e      
|   2          x                                                         |        
\  x                                                                     /

$$\left(\left(\cot^{2}{\left(x \right)} + 1\right) \left(\cot^{2}{\left(x \right)} + 2 \cot{\left(x \right)} + 1\right) \log{\left(4 x \right)} - \frac{2 \left(\cot^{2}{\left(x \right)} + 1\right)}{x} - \frac{1}{x^{2}}\right) e^{\cot{\left(x \right)}}$$

Simplify

The third derivative [src]

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

$$\left(- \left(\cot^{2}{\left(x \right)} + 1\right) \left(\left(\cot^{2}{\left(x \right)} + 1\right)^{2} + 6 \left(\cot^{2}{\left(x \right)} + 1\right) \cot{\left(x \right)} + 6 \cot^{2}{\left(x \right)} + 2\right) \log{\left(4 x \right)} + \frac{3 \left(\cot^{2}{\left(x \right)} + 1\right) \left(\cot^{2}{\left(x \right)} + 2 \cot{\left(x \right)} + 1\right)}{x} + \frac{3 \left(\cot^{2}{\left(x \right)} + 1\right)}{x^{2}} + \frac{2}{x^{3}}\right) e^{\cot{\left(x \right)}}$$

Simplify

The graph

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Identical expressions

Derivative of y=ln4x*e^ctgx

The graph:

Piecewise:

The solution

Method #1

Method #2