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Similar expressions
y=sqrt(1-cos^3x)

The derivative of the function/ y=sqrt(1+cos^3x)

Derivative of y=sqrt(1+cos^3x)

The solution

You have entered [src]

   _____________
  /        3    
\/  1 + cos (x)

$$\sqrt{\cos^{3}{\left(x \right)} + 1}$$

  /   _____________\
d |  /        3    |
--\\/  1 + cos (x) /
dx

$$\frac{d}{d x} \sqrt{\cos^{3}{\left(x \right)} + 1}$$

Transform

Detail solution

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

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

The answer is:

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

The graph

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Plot the graph f'(x)

The first derivative [src]

      2           
-3*cos (x)*sin(x) 
------------------
     _____________
    /        3    
2*\/  1 + cos (x)

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

  /                 2              5               6       2           3       2   \       
  |     2      7*cos (x)      9*cos (x)      27*cos (x)*sin (x)   9*cos (x)*sin (x)|       
3*|- sin (x) + --------- - --------------- - ------------------ + -----------------|*sin(x)
  |                2         /       3   \                   2       /       3   \ |       
  |                        4*\1 + cos (x)/      /       3   \      2*\1 + cos (x)/ |       
  \                                           8*\1 + cos (x)/                      /       
-------------------------------------------------------------------------------------------
                                         _____________                                     
                                        /        3                                         
                                      \/  1 + cos (x)

$$\frac{3 \left(- \sin^{2}{\left(x \right)} + \frac{7 \cos^{2}{\left(x \right)}}{2} + \frac{9 \sin^{2}{\left(x \right)} \cos^{3}{\left(x \right)}}{2 \left(\cos^{3}{\left(x \right)} + 1\right)} - \frac{9 \cos^{5}{\left(x \right)}}{4 \left(\cos^{3}{\left(x \right)} + 1\right)} - \frac{27 \sin^{2}{\left(x \right)} \cos^{6}{\left(x \right)}}{8 \left(\cos^{3}{\left(x \right)} + 1\right)^{2}}\right) \sin{\left(x \right)}}{\sqrt{\cos^{3}{\left(x \right)} + 1}}$$

Simplify

The graph

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Derivative of y=sqrt(1+cos^3x)

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The solution