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

Function f() - derivative -N order at the point
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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
  1. Apply the quotient rule, which is:

    and .

    To find :

    1. Let .

    2. The derivative of is itself.

    3. Then, apply the chain rule. Multiply by :

      1. The derivative of a constant times a function is the constant times the derivative of the function.

        1. Apply the power rule: goes to

        So, the result is:

      The result of the chain rule is:

    To find :

    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:

      2. Apply the quotient rule, which is:

        and .

        To find :

        1. Let .

        2. The derivative of sine is cosine:

        3. Then, apply the chain rule. Multiply by :

          1. The derivative of a constant times a function is the constant times the derivative of the function.

            1. Apply the power rule: goes to

            So, the result is:

          The result of the chain rule is:

        To find :

        1. Let .

        2. The derivative of cosine is negative sine:

        3. Then, apply the chain rule. Multiply by :

          1. The derivative of a constant times a function is the constant times the derivative of the function.

            1. Apply the power rule: goes to

            So, the result is:

          The result of the chain rule is:

        Now plug in to the quotient rule:

      So, the result is:

    Now plug in to the quotient rule:

  2. Now simplify:


The answer is:

The graph
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)      
$$\frac{\left(- 8 \tan^{2}{\left(4 x \right)} - 8\right) e^{4 x}}{4 \tan^{2}{\left(4 x \right)}} + 4 e^{4 x} \frac{1}{2 \tan{\left(4 x \right)}}$$
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)                               
$$\frac{8 \left(2 \left(\frac{\tan^{2}{\left(4 x \right)} + 1}{\tan^{2}{\left(4 x \right)}} - 1\right) \left(\tan^{2}{\left(4 x \right)} + 1\right) - \frac{2 \left(\tan^{2}{\left(4 x \right)} + 1\right)}{\tan{\left(4 x \right)}} + 1\right) e^{4 x}}{\tan{\left(4 x \right)}}$$
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)                                              /     
$$32 \left(\frac{6 \left(\frac{\tan^{2}{\left(4 x \right)} + 1}{\tan^{2}{\left(4 x \right)}} - 1\right) \left(\tan^{2}{\left(4 x \right)} + 1\right)}{\tan{\left(4 x \right)}} - \frac{6 \left(\tan^{2}{\left(4 x \right)} + 1\right)^{3}}{\tan^{4}{\left(4 x \right)}} + \frac{10 \left(\tan^{2}{\left(4 x \right)} + 1\right)^{2}}{\tan^{2}{\left(4 x \right)}} - \frac{3 \left(\tan^{2}{\left(4 x \right)} + 1\right)}{\tan^{2}{\left(4 x \right)}} - 4 \tan^{2}{\left(4 x \right)} - 4 + \frac{1}{\tan{\left(4 x \right)}}\right) e^{4 x}$$