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ln(tg3x+x^3)

Derivative of ln(tg3x+x^3)

Function f() - derivative -N order at the point
v

The graph:

from to

Piecewise:

The solution

You have entered [src]
   /            3\
log\tan(3*x) + x /
$$\log{\left(x^{3} + \tan{\left(3 x \right)} \right)}$$
d /   /            3\\
--\log\tan(3*x) + x //
dx                    
$$\frac{d}{d x} \log{\left(x^{3} + \tan{\left(3 x \right)} \right)}$$
Detail solution
  1. Let .

  2. The derivative of is .

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

    1. Differentiate term by term:

      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:

      3. Apply the power rule: goes to

      The result is:

    The result of the chain rule is:

  4. Now simplify:


The answer is:

The graph
The first derivative [src]
       2        2     
3 + 3*x  + 3*tan (3*x)
----------------------
                3     
    tan(3*x) + x      
$$\frac{3 x^{2} + 3 \tan^{2}{\left(3 x \right)} + 3}{x^{3} + \tan{\left(3 x \right)}}$$
The second derivative [src]
  /                            2                             \
  |        /     2      2     \                              |
  |      3*\1 + x  + tan (3*x)/      /       2     \         |
3*|2*x - ----------------------- + 6*\1 + tan (3*x)/*tan(3*x)|
  |            3                                             |
  \           x  + tan(3*x)                                  /
--------------------------------------------------------------
                         3                                    
                        x  + tan(3*x)                         
$$\frac{3 \cdot \left(2 x + 6 \left(\tan^{2}{\left(3 x \right)} + 1\right) \tan{\left(3 x \right)} - \frac{3 \left(x^{2} + \tan^{2}{\left(3 x \right)} + 1\right)^{2}}{x^{3} + \tan{\left(3 x \right)}}\right)}{x^{3} + \tan{\left(3 x \right)}}$$
The third derivative [src]
  /                                               3                                                                                         \
  |                     2     /     2      2     \                                     /      /       2     \         \ /     2      2     \|
  |      /       2     \    9*\1 + x  + tan (3*x)/          2      /       2     \   9*\x + 3*\1 + tan (3*x)/*tan(3*x)/*\1 + x  + tan (3*x)/|
6*|1 + 9*\1 + tan (3*x)/  + ----------------------- + 18*tan (3*x)*\1 + tan (3*x)/ - -------------------------------------------------------|
  |                                            2                                                           3                                |
  |                             / 3           \                                                           x  + tan(3*x)                     |
  \                             \x  + tan(3*x)/                                                                                             /
---------------------------------------------------------------------------------------------------------------------------------------------
                                                                 3                                                                           
                                                                x  + tan(3*x)                                                                
$$\frac{6 \left(- \frac{9 \left(x + 3 \left(\tan^{2}{\left(3 x \right)} + 1\right) \tan{\left(3 x \right)}\right) \left(x^{2} + \tan^{2}{\left(3 x \right)} + 1\right)}{x^{3} + \tan{\left(3 x \right)}} + 9 \left(\tan^{2}{\left(3 x \right)} + 1\right)^{2} + 18 \left(\tan^{2}{\left(3 x \right)} + 1\right) \tan^{2}{\left(3 x \right)} + 1 + \frac{9 \left(x^{2} + \tan^{2}{\left(3 x \right)} + 1\right)^{3}}{\left(x^{3} + \tan{\left(3 x \right)}\right)^{2}}\right)}{x^{3} + \tan{\left(3 x \right)}}$$
The graph
Derivative of ln(tg3x+x^3)