Mister Exam

Lang:

Other calculators

How to use it?
Derivative of:
Derivative of x-lnx
Derivative of (x^2-8*x)/(x+1)
Derivative of x^2+5
Derivative of x^2-4*x
Identical expressions
x*cos(two ^exp(x))*x^(seven -arccos(three))
x multiply by co sinus of e of (2 to the power of exponent of (x)) multiply by x to the power of (7 minus arc co sinus of e of (3))
x multiply by co sinus of e of (two to the power of exponent of (x)) multiply by x to the power of (seven minus arc co sinus of e of (three))
x*cos(2exp(x))*x(7-arccos(3))
x*cos2expx*x7-arccos3
xcos(2^exp(x))x^(7-arccos(3))
xcos(2exp(x))x(7-arccos(3))
xcos2expxx7-arccos3
xcos2^expxx^7-arccos3
Similar expressions
x*cos(2^exp(x))*x^(7+arccos(3))

The derivative of the function/ x*cos(2^exp(x))*x^(7-arccos(3))

Derivative of xcos(2^exp(x))x^(7-arccos(3))

The solution

You have entered [src]

     / / x\\             
     | \e /|  7 - acos(3)
x*cos\2    /*x

$$x^{7 - \operatorname{acos}{\left(3 \right)}} x \cos{\left(2^{e^{x}} \right)}$$

(x*cos(2^exp(x)))*x^(7 - acos(3))

Detail solution

Apply the product rule:

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

$f{\left(x \right)} = x \cos{\left(2^{e^{x}} \right)}$ ; to find $\frac{d}{d x} f{\left(x \right)}$ :
1. Apply the product rule:
  
  $\frac{d}{d x} f{\left(x \right)} g{\left(x \right)} = f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}$
  
  $f{\left(x \right)} = x$ ; to find $\frac{d}{d x} f{\left(x \right)}$ :
  1. Apply the power rule: $x$ goes to $1$
  $g{\left(x \right)} = \cos{\left(2^{e^{x}} \right)}$ ; to find $\frac{d}{d x} g{\left(x \right)}$ :
  1. Let $u = 2^{e^{x}}$ .
  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 $\frac{d}{d x} 2^{e^{x}}$ :
    1. Let $u = e^{x}$ .
    2. $\frac{d}{d u} 2^{u} = 2^{u} \log{\left(2 \right)}$
    3. Then, apply the chain rule. Multiply by $\frac{d}{d x} e^{x}$ :
      1. The derivative of $e^{x}$ is itself.
      The result of the chain rule is:
      
      $2^{e^{x}} e^{x} \log{\left(2 \right)}$
    The result of the chain rule is:
    
    $- 2^{e^{x}} e^{x} \log{\left(2 \right)} \sin{\left(2^{e^{x}} \right)}$
  The result is: $- 2^{e^{x}} x e^{x} \log{\left(2 \right)} \sin{\left(2^{e^{x}} \right)} + \cos{\left(2^{e^{x}} \right)}$
$g{\left(x \right)} = x^{7 - \operatorname{acos}{\left(3 \right)}}$ ; to find $\frac{d}{d x} g{\left(x \right)}$ :
1. Apply the power rule: $x^{7 - \operatorname{acos}{\left(3 \right)}}$ goes to $\frac{x^{7 - \operatorname{acos}{\left(3 \right)}} \left(7 - \operatorname{acos}{\left(3 \right)}\right)}{x}$
The result is: $x^{7 - \operatorname{acos}{\left(3 \right)}} \left(- 2^{e^{x}} x e^{x} \log{\left(2 \right)} \sin{\left(2^{e^{x}} \right)} + \cos{\left(2^{e^{x}} \right)}\right) + x^{7 - \operatorname{acos}{\left(3 \right)}} \left(7 - \operatorname{acos}{\left(3 \right)}\right) \cos{\left(2^{e^{x}} \right)}$
Now simplify:

$x^{7 - \operatorname{acos}{\left(3 \right)}} \left(- 2^{e^{x}} x e^{x} \log{\left(2 \right)} \sin{\left(2^{e^{x}} \right)} + 8 \cos{\left(2^{e^{x}} \right)} - \cos{\left(2^{e^{x}} \right)} \operatorname{acos}{\left(3 \right)}\right)$

The answer is:

$x^{7 - \operatorname{acos}{\left(3 \right)}} \left(- 2^{e^{x}} x e^{x} \log{\left(2 \right)} \sin{\left(2^{e^{x}} \right)} + 8 \cos{\left(2^{e^{x}} \right)} - \cos{\left(2^{e^{x}} \right)} \operatorname{acos}{\left(3 \right)}\right)$

The first derivative [src]

             /     / x\              / / x\\      / / x\\\                                 / / x\\
 7 - acos(3) |     \e /  x           | \e /|      | \e /||    7 - acos(3)                  | \e /|
x           *\- x*2    *e *log(2)*sin\2    / + cos\2    // + x           *(7 - acos(3))*cos\2    /

$$x^{7 - \operatorname{acos}{\left(3 \right)}} \left(- 2^{e^{x}} x e^{x} \log{\left(2 \right)} \sin{\left(2^{e^{x}} \right)} + \cos{\left(2^{e^{x}} \right)}\right) + x^{7 - \operatorname{acos}{\left(3 \right)}} \left(7 - \operatorname{acos}{\left(3 \right)}\right) \cos{\left(2^{e^{x}} \right)}$$

Simplify

The second derivative [src]

             /                  /     / / x\\      / x\              / / x\\\                                  / / x\\                                                                                                      \
             |                  |     | \e /|      \e /  x           | \e /||                                  | \e /|    / x\ /     / / x\\     /             / / x\\    / x\    / / x\\                / / x\\\\          |
 7 - acos(3) |  2*(7 - acos(3))*\- cos\2    / + x*2    *e *log(2)*sin\2    //   (6 - acos(3))*(7 - acos(3))*cos\2    /    \e / |     | \e /|     | x           | \e /|    \e /    | \e /|  x             | \e /|||  x       |
x           *|- ------------------------------------------------------------- + -------------------------------------- - 2    *\2*sin\2    / + x*\e *log(2)*sin\2    / + 2    *cos\2    /*e *log(2) + sin\2    ///*e *log(2)|
             \                                x                                                   x                                                                                                                         /

$$x^{7 - \operatorname{acos}{\left(3 \right)}} \left(- 2^{e^{x}} \left(x \left(2^{e^{x}} e^{x} \log{\left(2 \right)} \cos{\left(2^{e^{x}} \right)} + e^{x} \log{\left(2 \right)} \sin{\left(2^{e^{x}} \right)} + \sin{\left(2^{e^{x}} \right)}\right) + 2 \sin{\left(2^{e^{x}} \right)}\right) e^{x} \log{\left(2 \right)} - \frac{2 \left(7 - \operatorname{acos}{\left(3 \right)}\right) \left(2^{e^{x}} x e^{x} \log{\left(2 \right)} \sin{\left(2^{e^{x}} \right)} - \cos{\left(2^{e^{x}} \right)}\right)}{x} + \frac{\left(6 - \operatorname{acos}{\left(3 \right)}\right) \left(7 - \operatorname{acos}{\left(3 \right)}\right) \cos{\left(2^{e^{x}} \right)}}{x}\right)$$

Simplify

4-я производная [src]

             /                                                                       / / x\\                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                       /     / / x\\      / x\              / / x\\\                                         / x\               /     / / x\\     /                / / x\\                  / / x\\       x                 / / x\\      / x\            / / x\\           / x\    / / x\\                / / x\\\                  / / x\\      / x\    / / x\\          \                / x\                             /     / / x\\     /             / / x\\    / x\    / / x\\                / / x\\\\          \
             |              /                  3                               2\    | \e /|    / x\ /     / / x\\     /                / / x\\                     / / x\\                  / / x\\       x            / / x\\              x                 / / x\\         x                 / / x\\      / x\            / / x\\           / x\    / / x\\                 / x\            / / x\\           / / x\\\                     / / x\\                   / / x\\         x                 / / x\\       / x\            / / x\\            / x\    / / x\\          \                             |     | \e /|      \e /  x           | \e /|| /                   2            \      \e /               |     | \e /|     |   2     2*x    | \e /|      x           | \e /|    2*e     2     2*x    | \e /|      \e /    2       | \e /|  2*x      \e /    | \e /|  x             | \e /||      x           | \e /|      \e /    | \e /|  x       |  x             \e /                             |     | \e /|     | x           | \e /|    \e /    | \e /|  x             | \e /|||  x       |
 7 - acos(3) |(7 - acos(3))*\71 + (7 - acos(3))  - 11*acos(3) - 6*(7 - acos(3)) /*cos\2    /    \e / |     | \e /|     |   3     3*x    | \e /|        2     2*x    | \e /|      x           | \e /|    3*e     3       | \e /|  3*x      2*e     2     2*x    | \e /|      2*e     3     3*x    | \e /|      \e /    3       | \e /|  3*x      \e /    | \e /|  x              \e /    2       | \e /|  2*x      | \e /||        2     2*x    | \e /|       x           | \e /|      2*e     2     2*x    | \e /|       \e /    2       | \e /|  2*x       \e /    | \e /|  x       |  x          4*(7 - acos(3))*\- cos\2    / + x*2    *e *log(2)*sin\2    //*\-19 + (7 - acos(3))  + 3*acos(3)/   4*2    *(7 - acos(3))*\3*sin\2    / + x*\log (2)*e   *sin\2    / + 3*e *log(2)*sin\2    / - 2    *log (2)*e   *sin\2    / + 3*2    *log (2)*cos\2    /*e    + 3*2    *cos\2    /*e *log(2) + sin\2    // + 3*e *log(2)*sin\2    / + 3*2    *cos\2    /*e *log(2)/*e *log(2)   6*2    *(6 - acos(3))*(7 - acos(3))*\2*sin\2    / + x*\e *log(2)*sin\2    / + 2    *cos\2    /*e *log(2) + sin\2    ///*e *log(2)|
x           *|------------------------------------------------------------------------------ - 2    *\4*sin\2    / + x*\log (2)*e   *sin\2    / + 6*log (2)*e   *sin\2    / + 7*e *log(2)*sin\2    / - 2    *log (2)*cos\2    /*e    - 6*2    *log (2)*e   *sin\2    / - 6*2    *log (2)*e   *sin\2    / + 7*2    *log (2)*cos\2    /*e    + 7*2    *cos\2    /*e *log(2) + 18*2    *log (2)*cos\2    /*e    + sin\2    // + 4*log (2)*e   *sin\2    / + 12*e *log(2)*sin\2    / - 4*2    *log (2)*e   *sin\2    / + 12*2    *log (2)*cos\2    /*e    + 12*2    *cos\2    /*e *log(2)/*e *log(2) - ------------------------------------------------------------------------------------------------ - --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- - ---------------------------------------------------------------------------------------------------------------------------------|
             |                                       3                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                             3                                                                                                                                                                                       x                                                                                                                                                                                                         2                                                               |
             \                                      x                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                             x                                                                                                                                                                                                                                                                                                                                                                                                 x                                                                /

$$x^{7 - \operatorname{acos}{\left(3 \right)}} \left(- 2^{e^{x}} \left(- 4 \cdot 2^{2 e^{x}} e^{2 x} \log{\left(2 \right)}^{2} \sin{\left(2^{e^{x}} \right)} + 12 \cdot 2^{e^{x}} e^{2 x} \log{\left(2 \right)}^{2} \cos{\left(2^{e^{x}} \right)} + 12 \cdot 2^{e^{x}} e^{x} \log{\left(2 \right)} \cos{\left(2^{e^{x}} \right)} + x \left(- 2^{3 e^{x}} e^{3 x} \log{\left(2 \right)}^{3} \cos{\left(2^{e^{x}} \right)} - 6 \cdot 2^{2 e^{x}} e^{3 x} \log{\left(2 \right)}^{3} \sin{\left(2^{e^{x}} \right)} - 6 \cdot 2^{2 e^{x}} e^{2 x} \log{\left(2 \right)}^{2} \sin{\left(2^{e^{x}} \right)} + 7 \cdot 2^{e^{x}} e^{3 x} \log{\left(2 \right)}^{3} \cos{\left(2^{e^{x}} \right)} + 18 \cdot 2^{e^{x}} e^{2 x} \log{\left(2 \right)}^{2} \cos{\left(2^{e^{x}} \right)} + 7 \cdot 2^{e^{x}} e^{x} \log{\left(2 \right)} \cos{\left(2^{e^{x}} \right)} + e^{3 x} \log{\left(2 \right)}^{3} \sin{\left(2^{e^{x}} \right)} + 6 e^{2 x} \log{\left(2 \right)}^{2} \sin{\left(2^{e^{x}} \right)} + 7 e^{x} \log{\left(2 \right)} \sin{\left(2^{e^{x}} \right)} + \sin{\left(2^{e^{x}} \right)}\right) + 4 e^{2 x} \log{\left(2 \right)}^{2} \sin{\left(2^{e^{x}} \right)} + 12 e^{x} \log{\left(2 \right)} \sin{\left(2^{e^{x}} \right)} + 4 \sin{\left(2^{e^{x}} \right)}\right) e^{x} \log{\left(2 \right)} - \frac{4 \cdot 2^{e^{x}} \left(7 - \operatorname{acos}{\left(3 \right)}\right) \left(3 \cdot 2^{e^{x}} e^{x} \log{\left(2 \right)} \cos{\left(2^{e^{x}} \right)} + x \left(- 2^{2 e^{x}} e^{2 x} \log{\left(2 \right)}^{2} \sin{\left(2^{e^{x}} \right)} + 3 \cdot 2^{e^{x}} e^{2 x} \log{\left(2 \right)}^{2} \cos{\left(2^{e^{x}} \right)} + 3 \cdot 2^{e^{x}} e^{x} \log{\left(2 \right)} \cos{\left(2^{e^{x}} \right)} + e^{2 x} \log{\left(2 \right)}^{2} \sin{\left(2^{e^{x}} \right)} + 3 e^{x} \log{\left(2 \right)} \sin{\left(2^{e^{x}} \right)} + \sin{\left(2^{e^{x}} \right)}\right) + 3 e^{x} \log{\left(2 \right)} \sin{\left(2^{e^{x}} \right)} + 3 \sin{\left(2^{e^{x}} \right)}\right) e^{x} \log{\left(2 \right)}}{x} - \frac{6 \cdot 2^{e^{x}} \left(6 - \operatorname{acos}{\left(3 \right)}\right) \left(7 - \operatorname{acos}{\left(3 \right)}\right) \left(x \left(2^{e^{x}} e^{x} \log{\left(2 \right)} \cos{\left(2^{e^{x}} \right)} + e^{x} \log{\left(2 \right)} \sin{\left(2^{e^{x}} \right)} + \sin{\left(2^{e^{x}} \right)}\right) + 2 \sin{\left(2^{e^{x}} \right)}\right) e^{x} \log{\left(2 \right)}}{x^{2}} - \frac{4 \left(7 - \operatorname{acos}{\left(3 \right)}\right) \left(2^{e^{x}} x e^{x} \log{\left(2 \right)} \sin{\left(2^{e^{x}} \right)} - \cos{\left(2^{e^{x}} \right)}\right) \left(-19 + \left(7 - \operatorname{acos}{\left(3 \right)}\right)^{2} + 3 \operatorname{acos}{\left(3 \right)}\right)}{x^{3}} + \frac{\left(7 - \operatorname{acos}{\left(3 \right)}\right) \left(71 + \left(7 - \operatorname{acos}{\left(3 \right)}\right)^{3} - 11 \operatorname{acos}{\left(3 \right)} - 6 \left(7 - \operatorname{acos}{\left(3 \right)}\right)^{2}\right) \cos{\left(2^{e^{x}} \right)}}{x^{3}}\right)$$

Simplify

The third derivative [src]

             /                                                    / / x\\                                                                                                                                                                                                                                                                                               /     / / x\\      / x\              / / x\\\      / x\               /     / / x\\     /             / / x\\    / x\    / / x\\                / / x\\\\          \
             |              /                   2            \    | \e /|    / x\ /     / / x\\     /                / / x\\                  / / x\\       x                 / / x\\      / x\            / / x\\           / x\    / / x\\                / / x\\\                  / / x\\      / x\    / / x\\          \                                           |     | \e /|      \e /  x           | \e /||      \e /               |     | \e /|     | x           | \e /|    \e /    | \e /|  x             | \e /|||  x       |
 7 - acos(3) |(7 - acos(3))*\-19 + (7 - acos(3))  + 3*acos(3)/*cos\2    /    \e / |     | \e /|     |   2     2*x    | \e /|      x           | \e /|    2*e     2     2*x    | \e /|      \e /    2       | \e /|  2*x      \e /    | \e /|  x             | \e /||      x           | \e /|      \e /    | \e /|  x       |  x          3*(6 - acos(3))*(7 - acos(3))*\- cos\2    / + x*2    *e *log(2)*sin\2    //   3*2    *(7 - acos(3))*\2*sin\2    / + x*\e *log(2)*sin\2    / + 2    *cos\2    /*e *log(2) + sin\2    ///*e *log(2)|
x           *|----------------------------------------------------------- - 2    *\3*sin\2    / + x*\log (2)*e   *sin\2    / + 3*e *log(2)*sin\2    / - 2    *log (2)*e   *sin\2    / + 3*2    *log (2)*cos\2    /*e    + 3*2    *cos\2    /*e *log(2) + sin\2    // + 3*e *log(2)*sin\2    / + 3*2    *cos\2    /*e *log(2)/*e *log(2) - --------------------------------------------------------------------------- - -------------------------------------------------------------------------------------------------------------------|
             |                              2                                                                                                                                                                                                                                                                                                                                   2                                                                                                x                                                         |
             \                             x                                                                                                                                                                                                                                                                                                                                   x                                                                                                                                                           /

$$x^{7 - \operatorname{acos}{\left(3 \right)}} \left(- 2^{e^{x}} \left(3 \cdot 2^{e^{x}} e^{x} \log{\left(2 \right)} \cos{\left(2^{e^{x}} \right)} + x \left(- 2^{2 e^{x}} e^{2 x} \log{\left(2 \right)}^{2} \sin{\left(2^{e^{x}} \right)} + 3 \cdot 2^{e^{x}} e^{2 x} \log{\left(2 \right)}^{2} \cos{\left(2^{e^{x}} \right)} + 3 \cdot 2^{e^{x}} e^{x} \log{\left(2 \right)} \cos{\left(2^{e^{x}} \right)} + e^{2 x} \log{\left(2 \right)}^{2} \sin{\left(2^{e^{x}} \right)} + 3 e^{x} \log{\left(2 \right)} \sin{\left(2^{e^{x}} \right)} + \sin{\left(2^{e^{x}} \right)}\right) + 3 e^{x} \log{\left(2 \right)} \sin{\left(2^{e^{x}} \right)} + 3 \sin{\left(2^{e^{x}} \right)}\right) e^{x} \log{\left(2 \right)} - \frac{3 \cdot 2^{e^{x}} \left(7 - \operatorname{acos}{\left(3 \right)}\right) \left(x \left(2^{e^{x}} e^{x} \log{\left(2 \right)} \cos{\left(2^{e^{x}} \right)} + e^{x} \log{\left(2 \right)} \sin{\left(2^{e^{x}} \right)} + \sin{\left(2^{e^{x}} \right)}\right) + 2 \sin{\left(2^{e^{x}} \right)}\right) e^{x} \log{\left(2 \right)}}{x} - \frac{3 \left(6 - \operatorname{acos}{\left(3 \right)}\right) \left(7 - \operatorname{acos}{\left(3 \right)}\right) \left(2^{e^{x}} x e^{x} \log{\left(2 \right)} \sin{\left(2^{e^{x}} \right)} - \cos{\left(2^{e^{x}} \right)}\right)}{x^{2}} + \frac{\left(7 - \operatorname{acos}{\left(3 \right)}\right) \left(-19 + \left(7 - \operatorname{acos}{\left(3 \right)}\right)^{2} + 3 \operatorname{acos}{\left(3 \right)}\right) \cos{\left(2^{e^{x}} \right)}}{x^{2}}\right)$$

Simplify

Other calculators

How to use it?

Identical expressions

Similar expressions

Derivative of xcos(2^exp(x))x^(7-arccos(3))

The graph:

Piecewise:

The solution

Other calculators

How to use it?

Identical expressions

Similar expressions

Derivative of x*cos(2^exp(x))*x^(7-arccos(3))

The graph:

Piecewise:

The solution

Derivative of xcos(2^exp(x))x^(7-arccos(3))