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Identical expressions
y=xln((two -x)/(five +x))
y equally xln((2 minus x) divide by (5 plus x))
y equally xln((two minus x) divide by (five plus x))
y=xln2-x/5+x
y=xln((2-x) divide by (5+x))
Similar expressions
y=xln((2-x)/(5-x))
y=xln((2+x)/(5+x))

The derivative of the function/ y=xln((2-x)/(5+x))

Derivative of y=xln((2-x)/(5+x))

The solution

You have entered [src]

     /2 - x\
x*log|-----|
     \5 + x/

$$x \log{\left(\frac{2 - x}{x + 5} \right)}$$

x*log((2 - x)/(5 + x))

Detail solution

Apply the product rule:

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

$f{\left(x \right)} = x$ ; to find $\frac{d}{d x} f{\left(x \right)}$ :
1. Apply the power rule: $x$ goes to $1$
$g{\left(x \right)} = \log{\left(\frac{2 - x}{x + 5} \right)}$ ; to find $\frac{d}{d x} g{\left(x \right)}$ :
1. Let $u = \frac{2 - x}{x + 5}$ .
2. The derivative of $\log{\left(u \right)}$ is $\frac{1}{u}$ .
3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \frac{2 - x}{x + 5}$ :
  1. Apply the quotient rule, which is:
    
    $\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$
    
    $f{\left(x \right)} = 2 - x$ and $g{\left(x \right)} = x + 5$ .
    
    To find $\frac{d}{d x} f{\left(x \right)}$ :
    1. Differentiate $2 - x$ term by term:
      1. The derivative of the constant $2$ is zero.
      2. The derivative of a constant times a function is the constant times the derivative of the function.
        
        Apply the power rule: $x$ goes to $1$
        
        So, the result is: $-1$
      The result is: $-1$
    To find $\frac{d}{d x} g{\left(x \right)}$ :
    1. Differentiate $x + 5$ term by term:
      1. The derivative of the constant $5$ is zero.
      2. Apply the power rule: $x$ goes to $1$
      The result is: $1$
    Now plug in to the quotient rule:
    
    $- \frac{7}{\left(x + 5\right)^{2}}$
  The result of the chain rule is:
  
  $- \frac{7}{\left(2 - x\right) \left(x + 5\right)}$
The result is: $- \frac{7 x}{\left(2 - x\right) \left(x + 5\right)} + \log{\left(\frac{2 - x}{x + 5} \right)}$
Now simplify:

$\frac{7 x + \left(x - 2\right) \left(x + 5\right) \log{\left(\frac{2 - x}{x + 5} \right)}}{\left(x - 2\right) \left(x + 5\right)}$

The answer is:

$\frac{7 x + \left(x - 2\right) \left(x + 5\right) \log{\left(\frac{2 - x}{x + 5} \right)}}{\left(x - 2\right) \left(x + 5\right)}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

          /    1      2 - x  \             
x*(5 + x)*|- ----- - --------|             
          |  5 + x          2|             
          \          (5 + x) /      /2 - x\
------------------------------ + log|-----|
            2 - x                   \5 + x/

$$\frac{x \left(x + 5\right) \left(- \frac{2 - x}{\left(x + 5\right)^{2}} - \frac{1}{x + 5}\right)}{2 - x} + \log{\left(\frac{2 - x}{x + 5} \right)}$$

Simplify

The second derivative [src]

/     -2 + x\ /       /  1        1  \\
|-1 + ------|*|-2 + x*|------ + -----||
\     5 + x / \       \-2 + x   5 + x//
---------------------------------------
                 -2 + x

$$\frac{\left(x \left(\frac{1}{x + 5} + \frac{1}{x - 2}\right) - 2\right) \left(\frac{x - 2}{x + 5} - 1\right)}{x - 2}$$

Simplify

The third derivative [src]

/     -2 + x\ /  3        3         /    1          1              1        \\
|-1 + ------|*|------ + ----- - 2*x*|--------- + -------- + ----------------||
\     5 + x / |-2 + x   5 + x       |        2          2   (-2 + x)*(5 + x)||
              \                     \(-2 + x)    (5 + x)                    //
------------------------------------------------------------------------------
                                    -2 + x

$$\frac{\left(\frac{x - 2}{x + 5} - 1\right) \left(- 2 x \left(\frac{1}{\left(x + 5\right)^{2}} + \frac{1}{\left(x - 2\right) \left(x + 5\right)} + \frac{1}{\left(x - 2\right)^{2}}\right) + \frac{3}{x + 5} + \frac{3}{x - 2}\right)}{x - 2}$$

Simplify

The graph

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Derivative of y=xln((2-x)/(5+x))

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The solution