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y=ln(tg2x+1/4)

Derivative of y=ln(tg2x+1/4)

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

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The solution

You have entered [src]
log(tan(2*x) + 1/4)
$$\log{\left(\tan{\left(2 x \right)} + \frac{1}{4} \right)}$$
log(tan(2*x) + 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. 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. The derivative of the constant is zero.

      The result is:

    The result of the chain rule is:

  4. Now simplify:


The answer is:

The first derivative [src]
         2     
2 + 2*tan (2*x)
---------------
 tan(2*x) + 1/4
$$\frac{2 \tan^{2}{\left(2 x \right)} + 2}{\tan{\left(2 x \right)} + \frac{1}{4}}$$
The second derivative [src]
                   /    /       2     \           \
   /       2     \ |  2*\1 + tan (2*x)/           |
32*\1 + tan (2*x)/*|- ----------------- + tan(2*x)|
                   \    1 + 4*tan(2*x)            /
---------------------------------------------------
                   1 + 4*tan(2*x)                  
$$\frac{32 \left(\tan{\left(2 x \right)} - \frac{2 \left(\tan^{2}{\left(2 x \right)} + 1\right)}{4 \tan{\left(2 x \right)} + 1}\right) \left(\tan^{2}{\left(2 x \right)} + 1\right)}{4 \tan{\left(2 x \right)} + 1}$$
The third derivative [src]
                   /                                    2                              \
                   |                     /       2     \       /       2     \         |
   /       2     \ |         2        16*\1 + tan (2*x)/    12*\1 + tan (2*x)/*tan(2*x)|
64*\1 + tan (2*x)/*|1 + 3*tan (2*x) + ------------------- - ---------------------------|
                   |                                   2           1 + 4*tan(2*x)      |
                   \                   (1 + 4*tan(2*x))                                /
----------------------------------------------------------------------------------------
                                     1 + 4*tan(2*x)                                     
$$\frac{64 \left(\tan^{2}{\left(2 x \right)} + 1\right) \left(3 \tan^{2}{\left(2 x \right)} + 1 - \frac{12 \left(\tan^{2}{\left(2 x \right)} + 1\right) \tan{\left(2 x \right)}}{4 \tan{\left(2 x \right)} + 1} + \frac{16 \left(\tan^{2}{\left(2 x \right)} + 1\right)^{2}}{\left(4 \tan{\left(2 x \right)} + 1\right)^{2}}\right)}{4 \tan{\left(2 x \right)} + 1}$$
The graph
Derivative of y=ln(tg2x+1/4)