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Identical expressions
exp(four *x)/((two *tan(four *x)))
exponent of (4 multiply by x) divide by ((2 multiply by tangent of (4 multiply by x)))
exponent of (four multiply by x) divide by ((two multiply by tangent of (four multiply by x)))
exp(4x)/((2tan(4x)))
exp4x/2tan4x
exp(4*x) divide by ((2*tan(4*x)))

The derivative of the function/ exp(4*x)/((2*tan(4*x)))

Derivative of exp(4x)/((2tan(4*x)))

The solution

You have entered [src]

    4*x   
   e      
----------
2*tan(4*x)

$$\frac{e^{4 x}}{2 \tan{\left(4 x \right)}}$$

exp(4*x)/((2*tan(4*x)))

Detail solution

Apply the quotient rule, which is:

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

$f{\left(x \right)} = e^{4 x}$ and $g{\left(x \right)} = 2 \tan{\left(4 x \right)}$ .

To find $\frac{d}{d x} f{\left(x \right)}$ :
1. Let $u = 4 x$ .
2. The derivative of $e^{u}$ is itself.
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:
  
  $4 e^{4 x}$
To find $\frac{d}{d x} g{\left(x \right)}$ :
1. The derivative of a constant times a function is the constant times the derivative of the function.
  1. Rewrite the function to be differentiated:
    
    $\tan{\left(4 x \right)} = \frac{\sin{\left(4 x \right)}}{\cos{\left(4 x \right)}}$
  2. Apply the quotient rule, which is:
    
    $\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$
    
    $f{\left(x \right)} = \sin{\left(4 x \right)}$ and $g{\left(x \right)} = \cos{\left(4 x \right)}$ .
    
    To find $\frac{d}{d x} f{\left(x \right)}$ :
    1. Let $u = 4 x$ .
    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 $\frac{d}{d x} 4 x$ :
      1. 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: $4$
      The result of the chain rule is:
      
      $4 \cos{\left(4 x \right)}$
    To find $\frac{d}{d x} g{\left(x \right)}$ :
    1. Let $u = 4 x$ .
    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 $\frac{d}{d x} 4 x$ :
      1. 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: $4$
      The result of the chain rule is:
      
      $- 4 \sin{\left(4 x \right)}$
    Now plug in to the quotient rule:
    
    $\frac{4 \sin^{2}{\left(4 x \right)} + 4 \cos^{2}{\left(4 x \right)}}{\cos^{2}{\left(4 x \right)}}$
  So, the result is: $\frac{2 \left(4 \sin^{2}{\left(4 x \right)} + 4 \cos^{2}{\left(4 x \right)}\right)}{\cos^{2}{\left(4 x \right)}}$
Now plug in to the quotient rule:

$\frac{- \frac{2 \left(4 \sin^{2}{\left(4 x \right)} + 4 \cos^{2}{\left(4 x \right)}\right) e^{4 x}}{\cos^{2}{\left(4 x \right)}} + 8 e^{4 x} \tan{\left(4 x \right)}}{4 \tan^{2}{\left(4 x \right)}}$
Now simplify:

$\left(\frac{2}{\tan{\left(4 x \right)}} - \frac{2}{\sin^{2}{\left(4 x \right)}}\right) e^{4 x}$

The answer is:

$\left(\frac{2}{\tan{\left(4 x \right)}} - \frac{2}{\sin^{2}{\left(4 x \right)}}\right) e^{4 x}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

                    /          2     \  4*x
      1       4*x   \-8 - 8*tan (4*x)/*e   
4*----------*e    + -----------------------
  2*tan(4*x)                   2           
                          4*tan (4*x)

$$\frac{\left(- 8 \tan^{2}{\left(4 x \right)} - 8\right) e^{4 x}}{4 \tan^{2}{\left(4 x \right)}} + 4 e^{4 x} \frac{1}{2 \tan{\left(4 x \right)}}$$

Simplify

The second derivative [src]

  /      /       2     \                     /            2     \\     
  |    2*\1 + tan (4*x)/     /       2     \ |     1 + tan (4*x)||  4*x
8*|1 - ----------------- + 2*\1 + tan (4*x)/*|-1 + -------------||*e   
  |         tan(4*x)                         |          2       ||     
  \                                          \       tan (4*x)  //     
-----------------------------------------------------------------------
                                tan(4*x)

$$\frac{8 \left(2 \left(\frac{\tan^{2}{\left(4 x \right)} + 1}{\tan^{2}{\left(4 x \right)}} - 1\right) \left(\tan^{2}{\left(4 x \right)} + 1\right) - \frac{2 \left(\tan^{2}{\left(4 x \right)} + 1\right)}{\tan{\left(4 x \right)}} + 1\right) e^{4 x}}{\tan{\left(4 x \right)}}$$

Simplify

The third derivative [src]

   /                                                                                                               /            2     \\     
   |                                                                                               /       2     \ |     1 + tan (4*x)||     
   |                                               3                                         2   6*\1 + tan (4*x)/*|-1 + -------------||     
   |                                /       2     \      /       2     \      /       2     \                      |          2       ||     
   |        1            2        6*\1 + tan (4*x)/    3*\1 + tan (4*x)/   10*\1 + tan (4*x)/                      \       tan (4*x)  /|  4*x
32*|-4 + -------- - 4*tan (4*x) - ------------------ - ----------------- + ------------------- + --------------------------------------|*e   
   |     tan(4*x)                        4                    2                    2                            tan(4*x)               |     
   \                                  tan (4*x)            tan (4*x)            tan (4*x)                                              /

$$32 \left(\frac{6 \left(\frac{\tan^{2}{\left(4 x \right)} + 1}{\tan^{2}{\left(4 x \right)}} - 1\right) \left(\tan^{2}{\left(4 x \right)} + 1\right)}{\tan{\left(4 x \right)}} - \frac{6 \left(\tan^{2}{\left(4 x \right)} + 1\right)^{3}}{\tan^{4}{\left(4 x \right)}} + \frac{10 \left(\tan^{2}{\left(4 x \right)} + 1\right)^{2}}{\tan^{2}{\left(4 x \right)}} - \frac{3 \left(\tan^{2}{\left(4 x \right)} + 1\right)}{\tan^{2}{\left(4 x \right)}} - 4 \tan^{2}{\left(4 x \right)} - 4 + \frac{1}{\tan{\left(4 x \right)}}\right) e^{4 x}$$

Simplify

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Identical expressions

Derivative of exp(4x)/((2tan(4*x)))

The graph:

Piecewise:

The solution

Other calculators

How to use it?

Identical expressions

Derivative of exp(4*x)/((2*tan(4*x)))

The graph:

Piecewise:

The solution

Derivative of exp(4x)/((2tan(4*x)))