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Derivative of x^2*e^(-x)
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Derivative of -sin(x)
Identical expressions
one /(x/sqrt(x))-log(x)/(two /x^(three / two))
1 divide by (x divide by square root of (x)) minus logarithm of (x) divide by (2 divide by x to the power of (3 divide by 2))
one divide by (x divide by square root of (x)) minus logarithm of (x) divide by (two divide by x to the power of (three divide by two))
1/(x/√(x))-log(x)/(2/x^(3/2))
1/(x/sqrt(x))-log(x)/(2/x(3/2))
1/x/sqrtx-logx/2/x3/2
1/x/sqrtx-logx/2/x^3/2
1 divide by (x divide by sqrt(x))-log(x) divide by (2 divide by x^(3 divide by 2))
Similar expressions
1/(x/sqrt(x))+log(x)/(2/x^(3/2))

The derivative of the function/ 1/(x/sqrt(x))-log(x)/(2/x^(3/2))

Derivative of 1/(x/sqrt(x))-log(x)/(2/x^(3/2))

The solution

You have entered [src]

     1      log(x)
1*------- - ------
  /  x  \   / 2  \
  |-----|   |----|
  |  ___|   | 3/2|
  \\/ x /   \x   /

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

d /     1      log(x)\
--|1*------- - ------|
dx|  /  x  \   / 2  \|
  |  |-----|   |----||
  |  |  ___|   | 3/2||
  \  \\/ x /   \x   //

$$\frac{d}{d x} \left(- \frac{\log{\left(x \right)}}{2 \cdot \frac{1}{x^{\frac{3}{2}}}} + 1 \cdot \frac{1}{x \frac{1}{\sqrt{x}}}\right)$$

Transform

Detail solution

Differentiate $- \frac{\log{\left(x \right)}}{2 \cdot \frac{1}{x^{\frac{3}{2}}}} + 1 \cdot \frac{1}{x \frac{1}{\sqrt{x}}}$ term by term:
1. Apply the quotient rule, which is:
  
  $\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$
  
  $f{\left(x \right)} = \sqrt{x}$ and $g{\left(x \right)} = x$ .
  
  To find $\frac{d}{d x} f{\left(x \right)}$ :
  1. Apply the power rule: $\sqrt{x}$ goes to $\frac{1}{2 \sqrt{x}}$
  To find $\frac{d}{d x} g{\left(x \right)}$ :
  1. Apply the power rule: $x$ goes to $1$
  Now plug in to the quotient rule:
  
  $- \frac{1}{2 x^{\frac{3}{2}}}$
2. The derivative of a constant times a function is the constant times the derivative of the function.
  1. Apply the quotient rule, which is:
    
    $\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$
    
    $f{\left(x \right)} = x^{\frac{3}{2}} \log{\left(x \right)}$ and $g{\left(x \right)} = 2$ .
    
    To find $\frac{d}{d x} f{\left(x \right)}$ :
    1. Apply the product rule:
      
      $\frac{d}{d x} f{\left(x \right)} g{\left(x \right)} = f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}$
      
      $f{\left(x \right)} = x^{\frac{3}{2}}$ ; to find $\frac{d}{d x} f{\left(x \right)}$ :
      1. Apply the power rule: $x^{\frac{3}{2}}$ goes to $\frac{3 \sqrt{x}}{2}$
      $g{\left(x \right)} = \log{\left(x \right)}$ ; to find $\frac{d}{d x} g{\left(x \right)}$ :
      1. The derivative of $\log{\left(x \right)}$ is $\frac{1}{x}$ .
      The result is: $\frac{3 \sqrt{x} \log{\left(x \right)}}{2} + \sqrt{x}$
    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:
    
    $\frac{3 \sqrt{x} \log{\left(x \right)}}{4} + \frac{\sqrt{x}}{2}$
  So, the result is: $- \frac{3 \sqrt{x} \log{\left(x \right)}}{4} - \frac{\sqrt{x}}{2}$
The result is: $- \frac{3 \sqrt{x} \log{\left(x \right)}}{4} - \frac{\sqrt{x}}{2} - \frac{1}{2 x^{\frac{3}{2}}}$
Now simplify:

$\frac{x^{2} \left(- 3 \log{\left(x \right)} - 2\right) - 2}{4 x^{\frac{3}{2}}}$

The answer is:

$\frac{x^{2} \left(- 3 \log{\left(x \right)} - 2\right) - 2}{4 x^{\frac{3}{2}}}$

The graph

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Plot the graph f'(x)

The first derivative [src]

   1        1     / 3/2\                 
------- - -----   |x   |                 
    ___     ___   |----|       ___       
2*\/ x    \/ x    \ 2  /   3*\/ x *log(x)
--------------- - ------ - --------------
       x            x            4

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

Simplify

The second derivative [src]

     3*log(x)    3  
-1 - -------- + ----
        8          2
                4*x 
--------------------
         ___        
       \/ x

$$\frac{- \frac{3 \log{\left(x \right)}}{8} - 1 + \frac{3}{4 x^{2}}}{\sqrt{x}}$$

Simplify

The third derivative [src]

    30           
2 - -- + 3*log(x)
     2           
    x            
-----------------
         3/2     
     16*x

$$\frac{3 \log{\left(x \right)} + 2 - \frac{30}{x^{2}}}{16 x^{\frac{3}{2}}}$$

Simplify

The graph

Derivative of 1/(x/sqrt(x))-log(x)/(2/x^(3/2))

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Identical expressions

Similar expressions

Derivative of 1/(x/sqrt(x))-log(x)/(2/x^(3/2))

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The solution