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Identical expressions
sqrt(x^(three)-2x)cosx
square root of (x to the power of (3) minus 2x) co sinus of e of x
square root of (x to the power of (three) minus 2x) co sinus of e of x
√(x^(3)-2x)cosx
sqrt(x(3)-2x)cosx
sqrtx3-2xcosx
sqrtx^3-2xcosx
Similar expressions
sqrt(x^(3)+2x)cosx

The derivative of the function/ sqrt(x^(3)-2x)cosx

Derivative of sqrt(x^(3)-2x)cosx

The solution

You have entered [src]

   __________       
  /  3              
\/  x  - 2*x *cos(x)

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

sqrt(x^3 - 2*x)*cos(x)

Detail solution

Apply the product rule:

$\frac{d}{d x} f{\left(x \right)} g{\left(x \right)} = f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}$

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

                                                 /                  2\       
                                                 |       /        2\ |       
                                                 |       \-2 + 3*x / |       
                                                 |12*x - ------------|*cos(x)
     _____________          /        2\          |         /      2\ |       
    /   /      2\           \-2 + 3*x /*sin(x)   \       x*\-2 + x / /       
- \/  x*\-2 + x / *cos(x) - ------------------ + ----------------------------
                                _____________              _____________     
                               /   /      2\              /   /      2\      
                             \/  x*\-2 + x /          4*\/  x*\-2 + x /

$$- \sqrt{x \left(x^{2} - 2\right)} \cos{\left(x \right)} + \frac{\left(12 x - \frac{\left(3 x^{2} - 2\right)^{2}}{x \left(x^{2} - 2\right)}\right) \cos{\left(x \right)}}{4 \sqrt{x \left(x^{2} - 2\right)}} - \frac{\left(3 x^{2} - 2\right) \sin{\left(x \right)}}{\sqrt{x \left(x^{2} - 2\right)}}$$

Simplify

The third derivative [src]

                                                                                    /                                 3\       
                                                   /                  2\            |       /        2\    /        2\ |       
                                                   |       /        2\ |            |    12*\-2 + 3*x /    \-2 + 3*x / |       
                                                   |       \-2 + 3*x / |          3*|8 - -------------- + -------------|*cos(x)
                                                 3*|12*x - ------------|*sin(x)     |             2                   2|       
   _____________            /        2\            |         /      2\ |            |       -2 + x         2 /      2\ |       
  /   /      2\           3*\-2 + 3*x /*cos(x)     \       x*\-2 + x / /            \                     x *\-2 + x / /       
\/  x*\-2 + x / *sin(x) - -------------------- - ------------------------------ + ---------------------------------------------
                                _____________               _____________                            _____________             
                               /   /      2\               /   /      2\                            /   /      2\              
                           2*\/  x*\-2 + x /           4*\/  x*\-2 + x /                        8*\/  x*\-2 + x /

$$\sqrt{x \left(x^{2} - 2\right)} \sin{\left(x \right)} - \frac{3 \left(12 x - \frac{\left(3 x^{2} - 2\right)^{2}}{x \left(x^{2} - 2\right)}\right) \sin{\left(x \right)}}{4 \sqrt{x \left(x^{2} - 2\right)}} - \frac{3 \left(3 x^{2} - 2\right) \cos{\left(x \right)}}{2 \sqrt{x \left(x^{2} - 2\right)}} + \frac{3 \left(8 - \frac{12 \left(3 x^{2} - 2\right)}{x^{2} - 2} + \frac{\left(3 x^{2} - 2\right)^{3}}{x^{2} \left(x^{2} - 2\right)^{2}}\right) \cos{\left(x \right)}}{8 \sqrt{x \left(x^{2} - 2\right)}}$$

Simplify

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Derivative of sqrt(x^(3)-2x)cosx

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The solution