Mister Exam

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*x + 1\\
log|tan|-------||
   \   \   4   //
$$\log{\left(\tan{\left(\frac{2 x + 1}{4} \right)} \right)}$$
Detail solution
  1. Let .

  2. The derivative of is .

  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. 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. Differentiate term by term:

            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:

            2. The derivative of the constant is zero.

            The result is:

          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. Differentiate term by term:

            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:

            2. The derivative of the constant is zero.

            The result is:

          So, the result is:

        The result of the chain rule is:

      Now plug in to the quotient rule:

    The result of the chain rule is:

  4. Now simplify:


The answer is:

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