Mister Exam

Lang:

Other calculators

How to use it?
Derivative of:
Derivative of x/(x-4)
Derivative of x^3-3x^2+2
Derivative of |x-1|
Derivative of sqrt(cos(x))
Integral of d{x}:
(x^3cos(x/2)+1/2)sqrt(4-x^2)
Identical expressions
(x^3cos(x/ two)+ one / two)sqrt(four -x^ two)
(x cubed co sinus of e of (x divide by 2) plus 1 divide by 2) square root of (4 minus x squared )
(x cubed co sinus of e of (x divide by two) plus one divide by two) square root of (four minus x to the power of two)
(x^3cos(x/2)+1/2)√(4-x^2)
(x3cos(x/2)+1/2)sqrt(4-x2)
x3cosx/2+1/2sqrt4-x2
(x³cos(x/2)+1/2)sqrt(4-x²)
(x to the power of 3cos(x/2)+1/2)sqrt(4-x to the power of 2)
x^3cosx/2+1/2sqrt4-x^2
(x^3cos(x divide by 2)+1 divide by 2)sqrt(4-x^2)
Similar expressions
(x^3cos(x/2)-1/2)sqrt(4-x^2)
(x^3cos(x/2)+1/2)sqrt(4+x^2)

The derivative of the function/ (x^3cos(x/2)+1/2)sqrt(4-x^2)

Derivative of (x^3cos(x/2)+1/2)sqrt(4-x^2)

The solution

You have entered [src]

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

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

(x^3*cos(x/2) + 1/2)*sqrt(4 - x^2)

Detail solution

Apply the quotient rule, which is:

$\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$

$f{\left(x \right)} = \sqrt{4 - x^{2}} \left(2 x^{3} \cos{\left(\frac{x}{2} \right)} + 1\right)$ and $g{\left(x \right)} = 2$ .

To find $\frac{d}{d x} f{\left(x \right)}$ :
1. Apply the product rule:
  
  $\frac{d}{d x} f{\left(x \right)} g{\left(x \right)} = f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}$
  
  $f{\left(x \right)} = \sqrt{4 - x^{2}}$ ; to find $\frac{d}{d x} f{\left(x \right)}$ :
  1. Let $u = 4 - x^{2}$ .
  2. Apply the power rule: $\sqrt{u}$ goes to $\frac{1}{2 \sqrt{u}}$
  3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(4 - x^{2}\right)$ :
    1. Differentiate $4 - x^{2}$ term by term:
      1. The derivative of the constant $4$ is zero.
      2. The derivative of a constant times a function is the constant times the derivative of the function.
        
        Apply the power rule: $x^{2}$ goes to $2 x$
        
        So, the result is: $- 2 x$
      The result is: $- 2 x$
    The result of the chain rule is:
    
    $- \frac{x}{\sqrt{4 - x^{2}}}$
  $g{\left(x \right)} = 2 x^{3} \cos{\left(\frac{x}{2} \right)} + 1$ ; to find $\frac{d}{d x} g{\left(x \right)}$ :
  1. Differentiate $2 x^{3} \cos{\left(\frac{x}{2} \right)} + 1$ term by term:
    1. The derivative of the constant $1$ is zero.
    2. The derivative of a constant times a function is the constant times the derivative of the function.
      1. Apply the product rule:
        
        $\frac{d}{d x} f{\left(x \right)} g{\left(x \right)} = f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}$
        
        $f{\left(x \right)} = x^{3}$ ; to find $\frac{d}{d x} f{\left(x \right)}$ :
        
        Apply the power rule: $x^{3}$ goes to $3 x^{2}$
        
        $g{\left(x \right)} = \cos{\left(\frac{x}{2} \right)}$ ; to find $\frac{d}{d x} g{\left(x \right)}$ :
        
        Let $u = \frac{x}{2}$ .
        
        The derivative of cosine is negative sine:
        
        $\frac{d}{d u} \cos{\left(u \right)} = - \sin{\left(u \right)}$
        
        Then, apply the chain rule. Multiply by $\frac{d}{d x} \frac{x}{2}$ :
        
        The derivative of a constant times a function is the constant times the derivative of the function.
        
        Apply the power rule: $x$ goes to $1$
        
        So, the result is: $\frac{1}{2}$
        
        The result of the chain rule is:
        
        $- \frac{\sin{\left(\frac{x}{2} \right)}}{2}$
        
        The result is: $- \frac{x^{3} \sin{\left(\frac{x}{2} \right)}}{2} + 3 x^{2} \cos{\left(\frac{x}{2} \right)}$
      So, the result is: $- x^{3} \sin{\left(\frac{x}{2} \right)} + 6 x^{2} \cos{\left(\frac{x}{2} \right)}$
    The result is: $- x^{3} \sin{\left(\frac{x}{2} \right)} + 6 x^{2} \cos{\left(\frac{x}{2} \right)}$
  The result is: $- \frac{x \left(2 x^{3} \cos{\left(\frac{x}{2} \right)} + 1\right)}{\sqrt{4 - x^{2}}} + \sqrt{4 - x^{2}} \left(- x^{3} \sin{\left(\frac{x}{2} \right)} + 6 x^{2} \cos{\left(\frac{x}{2} \right)}\right)$
To find $\frac{d}{d x} g{\left(x \right)}$ :
1. The derivative of the constant $2$ is zero.
Now plug in to the quotient rule:

$- \frac{x \left(2 x^{3} \cos{\left(\frac{x}{2} \right)} + 1\right)}{2 \sqrt{4 - x^{2}}} + \frac{\sqrt{4 - x^{2}} \left(- x^{3} \sin{\left(\frac{x}{2} \right)} + 6 x^{2} \cos{\left(\frac{x}{2} \right)}\right)}{2}$
Now simplify:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

                                               /         2  \                                                        
                             /       3    /x\\ |        x   |        ________                                        
 3 /       /x\        /x\\   |1 + 2*x *cos|-||*|-1 + -------|       /      2  /        /x\    2    /x\           /x\\
x *|- 6*cos|-| + x*sin|-||   \            \2// |           2|   x*\/  4 - x  *|- 24*cos|-| + x *cos|-| + 12*x*sin|-||
   \       \2/        \2//                     \     -4 + x /                 \        \2/         \2/           \2//
-------------------------- + -------------------------------- - -----------------------------------------------------
          ________                         ________                                       4                          
         /      2                         /      2                                                                   
       \/  4 - x                      2*\/  4 - x

$$\frac{x^{3} \left(x \sin{\left(\frac{x}{2} \right)} - 6 \cos{\left(\frac{x}{2} \right)}\right)}{\sqrt{4 - x^{2}}} - \frac{x \sqrt{4 - x^{2}} \left(x^{2} \cos{\left(\frac{x}{2} \right)} + 12 x \sin{\left(\frac{x}{2} \right)} - 24 \cos{\left(\frac{x}{2} \right)}\right)}{4} + \frac{\left(\frac{x^{2}}{x^{2} - 4} - 1\right) \left(2 x^{3} \cos{\left(\frac{x}{2} \right)} + 1\right)}{2 \sqrt{4 - x^{2}}}$$

Simplify

The third derivative [src]

                                                                                                                        /         2  \                                                  /         2  \
                                                                                                                      2 |        x   | /       /x\        /x\\        /       3    /x\\ |        x   |
                                                                      2 /        /x\    2    /x\           /x\\   12*x *|-1 + -------|*|- 6*cos|-| + x*sin|-||   12*x*|1 + 2*x *cos|-||*|-1 + -------|
   ________                                                        6*x *|- 24*cos|-| + x *cos|-| + 12*x*sin|-||         |           2| \       \2/        \2//        \            \2// |           2|
  /      2  /      /x\    3    /x\           /x\       2    /x\\        \        \2/         \2/           \2//         \     -4 + x /                                                  \     -4 + x /
\/  4 - x  *|48*cos|-| + x *sin|-| - 72*x*sin|-| - 18*x *cos|-|| + -------------------------------------------- - -------------------------------------------- + -------------------------------------
            \      \2/         \2/           \2/            \2//                      ________                                       ________                                         3/2             
                                                                                     /      2                                       /      2                                  /     2\                
                                                                                   \/  4 - x                                      \/  4 - x                                   \4 - x /                
------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
                                                                                                  8

$$\frac{- \frac{12 x^{2} \left(x \sin{\left(\frac{x}{2} \right)} - 6 \cos{\left(\frac{x}{2} \right)}\right) \left(\frac{x^{2}}{x^{2} - 4} - 1\right)}{\sqrt{4 - x^{2}}} + \frac{6 x^{2} \left(x^{2} \cos{\left(\frac{x}{2} \right)} + 12 x \sin{\left(\frac{x}{2} \right)} - 24 \cos{\left(\frac{x}{2} \right)}\right)}{\sqrt{4 - x^{2}}} + \frac{12 x \left(\frac{x^{2}}{x^{2} - 4} - 1\right) \left(2 x^{3} \cos{\left(\frac{x}{2} \right)} + 1\right)}{\left(4 - x^{2}\right)^{\frac{3}{2}}} + \sqrt{4 - x^{2}} \left(x^{3} \sin{\left(\frac{x}{2} \right)} - 18 x^{2} \cos{\left(\frac{x}{2} \right)} - 72 x \sin{\left(\frac{x}{2} \right)} + 48 \cos{\left(\frac{x}{2} \right)}\right)}{8}$$

Simplify

Other calculators

How to use it?

Identical expressions

Similar expressions

Derivative of (x^3cos(x/2)+1/2)sqrt(4-x^2)

The graph:

Piecewise:

The solution