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(1+sin(6x))^1/4

Derivative of (1+sin(6x))^1/4

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

The graph:

from to

Piecewise:

The solution

You have entered [src]
4 ______________
\/ 1 + sin(6*x) 
$$\sqrt[4]{\sin{\left(6 x \right)} + 1}$$
d /4 ______________\
--\\/ 1 + sin(6*x) /
dx                  
$$\frac{d}{d x} \sqrt[4]{\sin{\left(6 x \right)} + 1}$$
Detail solution
  1. Let .

  2. Apply the power rule: goes to

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

    1. Differentiate term by term:

      1. The derivative of the constant is zero.

      2. Let .

      3. The derivative of sine is cosine:

      4. 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:

      The result is:

    The result of the chain rule is:


The answer is:

The graph
The first derivative [src]
     3*cos(6*x)    
-------------------
                3/4
2*(1 + sin(6*x))   
$$\frac{3 \cos{\left(6 x \right)}}{2 \left(\sin{\left(6 x \right)} + 1\right)^{\frac{3}{4}}}$$
The second derivative [src]
   /       2                   \
   |  3*cos (6*x)              |
-9*|---------------- + sin(6*x)|
   \4*(1 + sin(6*x))           /
--------------------------------
                     3/4        
       (1 + sin(6*x))           
$$- \frac{9 \left(\sin{\left(6 x \right)} + \frac{3 \cos^{2}{\left(6 x \right)}}{4 \left(\sin{\left(6 x \right)} + 1\right)}\right)}{\left(\sin{\left(6 x \right)} + 1\right)^{\frac{3}{4}}}$$
The third derivative [src]
   /                                 2       \         
   |        9*sin(6*x)         21*cos (6*x)  |         
27*|-2 + ---------------- + -----------------|*cos(6*x)
   |     2*(1 + sin(6*x))                   2|         
   \                        8*(1 + sin(6*x)) /         
-------------------------------------------------------
                                 3/4                   
                   (1 + sin(6*x))                      
$$\frac{27 \left(-2 + \frac{9 \sin{\left(6 x \right)}}{2 \left(\sin{\left(6 x \right)} + 1\right)} + \frac{21 \cos^{2}{\left(6 x \right)}}{8 \left(\sin{\left(6 x \right)} + 1\right)^{2}}\right) \cos{\left(6 x \right)}}{\left(\sin{\left(6 x \right)} + 1\right)^{\frac{3}{4}}}$$
The graph
Derivative of (1+sin(6x))^1/4