Derivative of sqrt(cos^3x)

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   _________
  /    3    
\/  cos (x)

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

sqrt(cos(x)^3)

Detail solution

Let $u = \cos^{3}{\left(x \right)}$ .
Apply the power rule: $\sqrt{u}$ goes to $\frac{1}{2 \sqrt{u}}$
Then, apply the chain rule. Multiply by $\frac{d}{d x} \cos^{3}{\left(x \right)}$ :
1. Let $u = \cos{\left(x \right)}$ .
2. Apply the power rule: $u^{3}$ goes to $3 u^{2}$
3. 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 of the chain rule is:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

      _________       
     /    3           
-3*\/  cos (x) *sin(x)
----------------------
       2*cos(x)

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

     _________ /        2   \       
    /    3     |     sin (x)|       
3*\/  cos (x) *|10 + -------|*sin(x)
               |        2   |       
               \     cos (x)/       
------------------------------------
              8*cos(x)

$$\frac{3 \left(\frac{\sin^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}} + 10\right) \sqrt{\cos^{3}{\left(x \right)}} \sin{\left(x \right)}}{8 \cos{\left(x \right)}}$$

Simplify

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Identical expressions

Derivative of sqrt(cos^3x)

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The solution