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sqrt(cos(x^ three + one))
square root of ( co sinus of e of (x cubed plus 1))
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√(cos(x^3+1))
sqrt(cos(x3+1))
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Similar expressions
sqrt(cos(x^3-1))

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

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

The solution

You have entered [src]

   _____________
  /    / 3    \ 
\/  cos\x  + 1/

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

sqrt(cos(x^3 + 1))

Detail solution

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

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

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

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

Simplify

The graph

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

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The solution