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(one +sin(6x))^ one / four
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(one plus sinus of (6x)) to the power of one divide by four
(1+sin(6x))1/4
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Similar expressions
(1-sin(6x))^1/4

The derivative of the function/ (1+sin(6x))^1/4

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

The solution

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4 ______________
\/ 1 + sin(6*x)

$$\sqrt[4]{\sin{\left(6 x \right)} + 1}$$

d /4 ______________\
--\\/ 1 + sin(6*x) /
dx

$$\frac{d}{d x} \sqrt[4]{\sin{\left(6 x \right)} + 1}$$

Transform

Detail solution

Let $u = \sin{\left(6 x \right)} + 1$ .
Apply the power rule: $\sqrt[4]{u}$ goes to $\frac{1}{4 u^{\frac{3}{4}}}$
Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(\sin{\left(6 x \right)} + 1\right)$ :
1. Differentiate $\sin{\left(6 x \right)} + 1$ term by term:
  1. The derivative of the constant $1$ is zero.
  2. Let $u = 6 x$ .
  3. The derivative of sine is cosine:
    
    $\frac{d}{d u} \sin{\left(u \right)} = \cos{\left(u \right)}$
  4. Then, apply the chain rule. Multiply by $\frac{d}{d x} 6 x$ :
    1. The derivative of a constant times a function is the constant times the derivative of the function.
      1. Apply the power rule: $x$ goes to $1$
      So, the result is: $6$
    The result of the chain rule is:
    
    $6 \cos{\left(6 x \right)}$
  The result is: $6 \cos{\left(6 x \right)}$
The result of the chain rule is:

$\frac{3 \cos{\left(6 x \right)}}{2 \left(\sin{\left(6 x \right)} + 1\right)^{\frac{3}{4}}}$

The answer is:

$\frac{3 \cos{\left(6 x \right)}}{2 \left(\sin{\left(6 x \right)} + 1\right)^{\frac{3}{4}}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

     3*cos(6*x)    
-------------------
                3/4
2*(1 + sin(6*x))

$$\frac{3 \cos{\left(6 x \right)}}{2 \left(\sin{\left(6 x \right)} + 1\right)^{\frac{3}{4}}}$$

Simplify

The second derivative [src]

   /       2                   \
   |  3*cos (6*x)              |
-9*|---------------- + sin(6*x)|
   \4*(1 + sin(6*x))           /
--------------------------------
                     3/4        
       (1 + sin(6*x))

$$- \frac{9 \left(\sin{\left(6 x \right)} + \frac{3 \cos^{2}{\left(6 x \right)}}{4 \left(\sin{\left(6 x \right)} + 1\right)}\right)}{\left(\sin{\left(6 x \right)} + 1\right)^{\frac{3}{4}}}$$

Simplify

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))

$$\frac{27 \left(-2 + \frac{9 \sin{\left(6 x \right)}}{2 \left(\sin{\left(6 x \right)} + 1\right)} + \frac{21 \cos^{2}{\left(6 x \right)}}{8 \left(\sin{\left(6 x \right)} + 1\right)^{2}}\right) \cos{\left(6 x \right)}}{\left(\sin{\left(6 x \right)} + 1\right)^{\frac{3}{4}}}$$

Simplify

The graph

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

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Piecewise:

The solution