Mister Exam

Derivative of lntg(2x+1)/4

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

The graph:

from to

Piecewise:

The solution

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

    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. 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:

          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. 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:

          The result of the chain rule is:

        Now plug in to the quotient rule:

      The result of the chain rule is:

    So, the result is:

  2. Now simplify:


The answer is:

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