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Derivative of:
Derivative of tanh(x)
Derivative of tanx
Derivative of tg2x
Derivative of (-1-x^2)/x
Identical expressions
sqrt^ three (x)cosx
square root of cubed (x) co sinus of e of x
square root of to the power of three (x) co sinus of e of x
√^3(x)cosx
sqrt3(x)cosx
sqrt3xcosx
sqrt³(x)cosx
sqrt to the power of 3(x)cosx
sqrt^3xcosx

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

Derivative of sqrt^3(x)cosx

The solution

You have entered [src]

     3       
  ___        
\/ x  *cos(x)

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

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

Detail solution

Apply the product rule:

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

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

                  ___                             
 3/2          9*\/ x *cos(x)   9*sin(x)   3*cos(x)
x   *sin(x) - -------------- - -------- - --------
                    2              ___        3/2 
                               4*\/ x      8*x

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

Simplify

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Identical expressions

Derivative of sqrt^3(x)cosx

The graph:

Piecewise:

The solution