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Derivative of f(x)=5
Identical expressions
x*sqrt(ln(5x- eight))
x multiply by square root of (ln(5x minus 8))
x multiply by square root of (ln(5x minus eight))
x*√(ln(5x-8))
xsqrt(ln(5x-8))
xsqrtln5x-8
Similar expressions
x*sqrt(ln(5x+8))

The derivative of the function/ x*sqrt(ln(5x-8))

Derivative of x*sqrt(ln(5x-8))

The solution

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    ______________
x*\/ log(5*x - 8)

$$x \sqrt{\log{\left(5 x - 8 \right)}}$$

x*sqrt(log(5*x - 8))

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 $1$
$g{\left(x \right)} = \sqrt{\log{\left(5 x - 8 \right)}}$ ; to find $\frac{d}{d x} g{\left(x \right)}$ :
1. Let $u = \log{\left(5 x - 8 \right)}$ .
2. Apply the power rule: $\sqrt{u}$ goes to $\frac{1}{2 \sqrt{u}}$
3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \log{\left(5 x - 8 \right)}$ :
  1. Let $u = 5 x - 8$ .
  2. The derivative of $\log{\left(u \right)}$ is $\frac{1}{u}$ .
  3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(5 x - 8\right)$ :
    1. Differentiate $5 x - 8$ term by term:
      1. 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: $5$
      2. The derivative of the constant $-8$ is zero.
      The result is: $5$
    The result of the chain rule is:
    
    $\frac{5}{5 x - 8}$
  The result of the chain rule is:
  
  $\frac{5}{2 \left(5 x - 8\right) \sqrt{\log{\left(5 x - 8 \right)}}}$
The result is: $\frac{5 x}{2 \left(5 x - 8\right) \sqrt{\log{\left(5 x - 8 \right)}}} + \sqrt{\log{\left(5 x - 8 \right)}}$
Now simplify:

$\frac{\frac{5 x}{2} + \left(5 x - 8\right) \log{\left(5 x - 8 \right)}}{\left(5 x - 8\right) \sqrt{\log{\left(5 x - 8 \right)}}}$

The answer is:

$\frac{\frac{5 x}{2} + \left(5 x - 8\right) \log{\left(5 x - 8 \right)}}{\left(5 x - 8\right) \sqrt{\log{\left(5 x - 8 \right)}}}$

The graph

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The first derivative [src]

  ______________               5*x             
\/ log(5*x - 8)  + ----------------------------
                                 ______________
                   2*(5*x - 8)*\/ log(5*x - 8)

$$\frac{5 x}{2 \left(5 x - 8\right) \sqrt{\log{\left(5 x - 8 \right)}}} + \sqrt{\log{\left(5 x - 8 \right)}}$$

Simplify

The second derivative [src]

  /        /          1      \\
  |    5*x*|2 + -------------||
  |        \    log(-8 + 5*x)/|
5*|1 - -----------------------|
  \          4*(-8 + 5*x)     /
-------------------------------
               _______________ 
  (-8 + 5*x)*\/ log(-8 + 5*x)

$$\frac{5 \left(- \frac{5 x \left(2 + \frac{1}{\log{\left(5 x - 8 \right)}}\right)}{4 \left(5 x - 8\right)} + 1\right)}{\left(5 x - 8\right) \sqrt{\log{\left(5 x - 8 \right)}}}$$

Simplify

The third derivative [src]

   /                          /          3                6      \\
   |                      5*x*|8 + -------------- + -------------||
   |                          |       2             log(-8 + 5*x)||
   |            6             \    log (-8 + 5*x)                /|
25*|-12 - ------------- + ----------------------------------------|
   \      log(-8 + 5*x)                   -8 + 5*x                /
-------------------------------------------------------------------
                              2   _______________                  
                  8*(-8 + 5*x) *\/ log(-8 + 5*x)

$$\frac{25 \left(\frac{5 x \left(8 + \frac{6}{\log{\left(5 x - 8 \right)}} + \frac{3}{\log{\left(5 x - 8 \right)}^{2}}\right)}{5 x - 8} - 12 - \frac{6}{\log{\left(5 x - 8 \right)}}\right)}{8 \left(5 x - 8\right)^{2} \sqrt{\log{\left(5 x - 8 \right)}}}$$

Simplify

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Derivative of x*sqrt(ln(5x-8))

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The solution