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y=ln(tg^2x+1/4)

Derivative of y=ln(tg^2x+1/4)

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

The graph:

from to

Piecewise:

The solution

You have entered [src]
   /   2      1\
log|tan (x) + -|
   \          4/
$$\log{\left(\tan^{2}{\left(x \right)} + \frac{1}{4} \right)}$$
log(tan(x)^2 + 1/4)
Detail solution
  1. Let .

  2. The derivative of is .

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

    1. Differentiate term by term:

      1. Let .

      2. Apply the power rule: goes to

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

        1. Rewrite the function to be differentiated:

        2. Apply the quotient rule, which is:

          and .

          To find :

          1. The derivative of sine is cosine:

          To find :

          1. The derivative of cosine is negative sine:

          Now plug in to the quotient rule:

        The result of the chain rule is:

      4. The derivative of the constant is zero.

      The result is:

    The result of the chain rule is:

  4. Now simplify:


The answer is:

The graph
The first derivative [src]
/         2   \       
\2 + 2*tan (x)/*tan(x)
----------------------
        2      1      
     tan (x) + -      
               4      
$$\frac{\left(2 \tan^{2}{\left(x \right)} + 2\right) \tan{\left(x \right)}}{\tan^{2}{\left(x \right)} + \frac{1}{4}}$$
The second derivative [src]
                /                     2    /       2   \\
  /       2   \ |         2      8*tan (x)*\1 + tan (x)/|
8*\1 + tan (x)/*|1 + 3*tan (x) - -----------------------|
                |                              2        |
                \                     1 + 4*tan (x)     /
---------------------------------------------------------
                               2                         
                      1 + 4*tan (x)                      
$$\frac{8 \left(\tan^{2}{\left(x \right)} + 1\right) \left(- \frac{8 \left(\tan^{2}{\left(x \right)} + 1\right) \tan^{2}{\left(x \right)}}{4 \tan^{2}{\left(x \right)} + 1} + 3 \tan^{2}{\left(x \right)} + 1\right)}{4 \tan^{2}{\left(x \right)} + 1}$$
The third derivative [src]
                 /                               2                                              2        \       
                 |                  /       2   \          2    /       2   \      /       2   \     2   |       
   /       2   \ |         2      6*\1 + tan (x)/    12*tan (x)*\1 + tan (x)/   32*\1 + tan (x)/ *tan (x)|       
32*\1 + tan (x)/*|2 + 3*tan (x) - ---------------- - ------------------------ + -------------------------|*tan(x)
                 |                          2                      2                                2    |       
                 |                 1 + 4*tan (x)          1 + 4*tan (x)              /         2   \     |       
                 \                                                                   \1 + 4*tan (x)/     /       
-----------------------------------------------------------------------------------------------------------------
                                                           2                                                     
                                                  1 + 4*tan (x)                                                  
$$\frac{32 \left(\tan^{2}{\left(x \right)} + 1\right) \left(- \frac{6 \left(\tan^{2}{\left(x \right)} + 1\right)^{2}}{4 \tan^{2}{\left(x \right)} + 1} + \frac{32 \left(\tan^{2}{\left(x \right)} + 1\right)^{2} \tan^{2}{\left(x \right)}}{\left(4 \tan^{2}{\left(x \right)} + 1\right)^{2}} - \frac{12 \left(\tan^{2}{\left(x \right)} + 1\right) \tan^{2}{\left(x \right)}}{4 \tan^{2}{\left(x \right)} + 1} + 3 \tan^{2}{\left(x \right)} + 2\right) \tan{\left(x \right)}}{4 \tan^{2}{\left(x \right)} + 1}$$
The graph
Derivative of y=ln(tg^2x+1/4)