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Derivative of 6
Derivative of x^7*e^x
Derivative of x/3+3/x
Derivative of 6^x
Identical expressions
y=(x^ five - two)*sinx/lnxe^x
y equally (x to the power of 5 minus 2) multiply by sinus of x divide by lnxe to the power of x
y equally (x to the power of five minus two) multiply by sinus of x divide by lnxe to the power of x
y=(x5-2)*sinx/lnxex
y=x5-2*sinx/lnxex
y=(x⁵-2)*sinx/lnxe^x
y=(x^5-2)sinx/lnxe^x
y=(x5-2)sinx/lnxex
y=x5-2sinx/lnxex
y=x^5-2sinx/lnxe^x
y=(x^5-2)*sinx divide by lnxe^x
Similar expressions
y=(x^5+2)*sinx/lnxe^x

The derivative of the function/ y=(x^5-2)*sinx/lnxe^x

Derivative of y=(x^5-2)*sinx/lnxe^x

The solution

You have entered [src]

/ 5    \          
\x  - 2/*sin(x)  x
---------------*E 
     log(x)

$$e^{x} \frac{\left(x^{5} - 2\right) \sin{\left(x \right)}}{\log{\left(x \right)}}$$

(((x^5 - 2)*sin(x))/log(x))*E^x

Detail solution

Apply the quotient rule, which is:

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

$f{\left(x \right)} = \left(x^{5} - 2\right) e^{x} \sin{\left(x \right)}$ and $g{\left(x \right)} = \log{\left(x \right)}$ .

To find $\frac{d}{d x} f{\left(x \right)}$ :
1. Apply the product rule:
  
  $\frac{d}{d x} f{\left(x \right)} g{\left(x \right)} h{\left(x \right)} = f{\left(x \right)} g{\left(x \right)} \frac{d}{d x} h{\left(x \right)} + f{\left(x \right)} h{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} h{\left(x \right)} \frac{d}{d x} f{\left(x \right)}$
  
  $f{\left(x \right)} = x^{5} - 2$ ; to find $\frac{d}{d x} f{\left(x \right)}$ :
  1. Differentiate $x^{5} - 2$ term by term:
    1. The derivative of the constant $-2$ is zero.
    2. Apply the power rule: $x^{5}$ goes to $5 x^{4}$
    The result is: $5 x^{4}$
  $g{\left(x \right)} = e^{x}$ ; to find $\frac{d}{d x} g{\left(x \right)}$ :
  1. The derivative of $e^{x}$ is itself.
  $h{\left(x \right)} = \sin{\left(x \right)}$ ; to find $\frac{d}{d x} h{\left(x \right)}$ :
  1. The derivative of sine is cosine:
    
    $\frac{d}{d x} \sin{\left(x \right)} = \cos{\left(x \right)}$
  The result is: $5 x^{4} e^{x} \sin{\left(x \right)} + \left(x^{5} - 2\right) e^{x} \sin{\left(x \right)} + \left(x^{5} - 2\right) e^{x} \cos{\left(x \right)}$
To find $\frac{d}{d x} g{\left(x \right)}$ :
1. The derivative of $\log{\left(x \right)}$ is $\frac{1}{x}$ .
Now plug in to the quotient rule:

$\frac{\left(5 x^{4} e^{x} \sin{\left(x \right)} + \left(x^{5} - 2\right) e^{x} \sin{\left(x \right)} + \left(x^{5} - 2\right) e^{x} \cos{\left(x \right)}\right) \log{\left(x \right)} - \frac{\left(x^{5} - 2\right) e^{x} \sin{\left(x \right)}}{x}}{\log{\left(x \right)}^{2}}$
Now simplify:

$\frac{\left(x \left(5 x^{4} \sin{\left(x \right)} + \left(x^{5} - 2\right) \sin{\left(x \right)} + \left(x^{5} - 2\right) \cos{\left(x \right)}\right) \log{\left(x \right)} + \left(2 - x^{5}\right) \sin{\left(x \right)}\right) e^{x}}{x \log{\left(x \right)}^{2}}$

The answer is:

$\frac{\left(x \left(5 x^{4} \sin{\left(x \right)} + \left(x^{5} - 2\right) \sin{\left(x \right)} + \left(x^{5} - 2\right) \cos{\left(x \right)}\right) \log{\left(x \right)} + \left(2 - x^{5}\right) \sin{\left(x \right)}\right) e^{x}}{x \log{\left(x \right)}^{2}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

// 5    \             4          / 5    \       \      / 5    \  x       
|\x  - 2/*cos(x) + 5*x *sin(x)   \x  - 2/*sin(x)|  x   \x  - 2/*e *sin(x)
|----------------------------- - ---------------|*e  + ------------------
|            log(x)                      2      |            log(x)      
\                                   x*log (x)   /

$$\frac{\left(x^{5} - 2\right) e^{x} \sin{\left(x \right)}}{\log{\left(x \right)}} + \left(\frac{5 x^{4} \sin{\left(x \right)} + \left(x^{5} - 2\right) \cos{\left(x \right)}}{\log{\left(x \right)}} - \frac{\left(x^{5} - 2\right) \sin{\left(x \right)}}{x \log{\left(x \right)}^{2}}\right) e^{x}$$

Simplify

The second derivative [src]

/                                                                                                                            /      2   \ /      5\       \   
|                                                                    //      5\             4       \     /      5\          |1 + ------|*\-2 + x /*sin(x)|   
|  /      5\              4              4              3          2*\\-2 + x /*cos(x) + 5*x *sin(x)/   2*\-2 + x /*sin(x)   \    log(x)/                 |  x
|2*\-2 + x /*cos(x) + 10*x *cos(x) + 10*x *sin(x) + 20*x *sin(x) - ---------------------------------- - ------------------ + -----------------------------|*e 
|                                                                               x*log(x)                     x*log(x)                   2                 |   
\                                                                                                                                      x *log(x)          /   
--------------------------------------------------------------------------------------------------------------------------------------------------------------
                                                                            log(x)

$$\frac{\left(10 x^{4} \sin{\left(x \right)} + 10 x^{4} \cos{\left(x \right)} + 20 x^{3} \sin{\left(x \right)} + 2 \left(x^{5} - 2\right) \cos{\left(x \right)} - \frac{2 \left(x^{5} - 2\right) \sin{\left(x \right)}}{x \log{\left(x \right)}} - \frac{2 \left(5 x^{4} \sin{\left(x \right)} + \left(x^{5} - 2\right) \cos{\left(x \right)}\right)}{x \log{\left(x \right)}} + \frac{\left(1 + \frac{2}{\log{\left(x \right)}}\right) \left(x^{5} - 2\right) \sin{\left(x \right)}}{x^{2} \log{\left(x \right)}}\right) e^{x}}{\log{\left(x \right)}}$$

Simplify

The third derivative [src]

/                                                                                                                                                                                                                                                                             /      5\ /      3         3   \                                         \   
|                                                                                                                                                                                                                           /      2   \ //      5\             4       \   2*\-2 + x /*|1 + ------ + -------|*sin(x)     /      2   \ /      5\       |   
|                                                                                                          //      5\             4       \     /  /      5\              4              3       \     /      5\          3*|1 + ------|*\\-2 + x /*cos(x) + 5*x *sin(x)/               |    log(x)      2   |          3*|1 + ------|*\-2 + x /*sin(x)|   
|    /      5\            /      5\              4              2              3              3          6*\\-2 + x /*cos(x) + 5*x *sin(x)/   3*\- \-2 + x /*sin(x) + 10*x *cos(x) + 20*x *sin(x)/   3*\-2 + x /*sin(x)     \    log(x)/                                                \             log (x)/            \    log(x)/                 |  x
|- 2*\-2 + x /*sin(x) + 2*\-2 + x /*cos(x) + 30*x *cos(x) + 60*x *sin(x) + 60*x *cos(x) + 60*x *sin(x) - ---------------------------------- - ---------------------------------------------------- - ------------------ + ----------------------------------------------- - ----------------------------------------- + -------------------------------|*e 
|                                                                                                                     x*log(x)                                      x*log(x)                              x*log(x)                            2                                              3                                      2                  |   
\                                                                                                                                                                                                                                            x *log(x)                                      x *log(x)                              x *log(x)           /   
-----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
                                                                                                                                                                           log(x)

$$\frac{\left(30 x^{4} \cos{\left(x \right)} + 60 x^{3} \sin{\left(x \right)} + 60 x^{3} \cos{\left(x \right)} + 60 x^{2} \sin{\left(x \right)} - 2 \left(x^{5} - 2\right) \sin{\left(x \right)} + 2 \left(x^{5} - 2\right) \cos{\left(x \right)} - \frac{3 \left(x^{5} - 2\right) \sin{\left(x \right)}}{x \log{\left(x \right)}} - \frac{6 \left(5 x^{4} \sin{\left(x \right)} + \left(x^{5} - 2\right) \cos{\left(x \right)}\right)}{x \log{\left(x \right)}} - \frac{3 \left(10 x^{4} \cos{\left(x \right)} + 20 x^{3} \sin{\left(x \right)} - \left(x^{5} - 2\right) \sin{\left(x \right)}\right)}{x \log{\left(x \right)}} + \frac{3 \left(1 + \frac{2}{\log{\left(x \right)}}\right) \left(x^{5} - 2\right) \sin{\left(x \right)}}{x^{2} \log{\left(x \right)}} + \frac{3 \left(1 + \frac{2}{\log{\left(x \right)}}\right) \left(5 x^{4} \sin{\left(x \right)} + \left(x^{5} - 2\right) \cos{\left(x \right)}\right)}{x^{2} \log{\left(x \right)}} - \frac{2 \left(x^{5} - 2\right) \left(1 + \frac{3}{\log{\left(x \right)}} + \frac{3}{\log{\left(x \right)}^{2}}\right) \sin{\left(x \right)}}{x^{3} \log{\left(x \right)}}\right) e^{x}}{\log{\left(x \right)}}$$

Simplify

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Derivative of y=(x^5-2)*sinx/lnxe^x

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The solution