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Derivative of (5x^2+2)×(x^2-4)^3
Identical expressions
sqrt(two log(5x+ one ,2))
square root of (2 logarithm of (5x plus 1,2))
square root of (two logarithm of (5x plus one ,2))
√(2log(5x+1,2))
sqrt2log5x+1,2
Similar expressions
sqrt(2log(5x-1,2))

The derivative of the function/ sqrt(2log(5x+1,2))

Derivative of sqrt(2log(5x+1,2))

The solution

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  __________________
\/ 2*log(5*x + 6/5)

$$\sqrt{2 \log{\left(5 x + \frac{6}{5} \right)}}$$

sqrt(2*log(5*x + 6/5))

Detail solution

Let $u = 2 \log{\left(5 x + \frac{6}{5} \right)}$ .
Apply the power rule: $\sqrt{u}$ goes to $\frac{1}{2 \sqrt{u}}$
Then, apply the chain rule. Multiply by $\frac{d}{d x} 2 \log{\left(5 x + \frac{6}{5} \right)}$ :
1. The derivative of a constant times a function is the constant times the derivative of the function.
  1. Let $u = 5 x + \frac{6}{5}$ .
  2. The derivative of $\log{\left(u \right)}$ is $\frac{1}{u}$ .
  3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(5 x + \frac{6}{5}\right)$ :
    1. Differentiate $5 x + \frac{6}{5}$ 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 $\frac{6}{5}$ is zero.
      The result is: $5$
    The result of the chain rule is:
    
    $\frac{5}{5 x + \frac{6}{5}}$
  So, the result is: $\frac{10}{5 x + \frac{6}{5}}$
The result of the chain rule is:

$\frac{5 \sqrt{2}}{2 \left(5 x + \frac{6}{5}\right) \sqrt{\log{\left(5 x + \frac{6}{5} \right)}}}$
Now simplify:

$\frac{25 \sqrt{2}}{2 \left(25 x + 6\right) \sqrt{\log{\left(5 x + \frac{6}{5} \right)}}}$

The answer is:

$\frac{25 \sqrt{2}}{2 \left(25 x + 6\right) \sqrt{\log{\left(5 x + \frac{6}{5} \right)}}}$

The graph

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The first derivative [src]

     ___   ________________ 
 5*\/ 2 *\/ log(5*x + 6/5)  
----------------------------
2*(5*x + 6/5)*log(5*x + 6/5)

$$\frac{5 \sqrt{2} \sqrt{\log{\left(5 x + \frac{6}{5} \right)}}}{2 \left(5 x + \frac{6}{5}\right) \log{\left(5 x + \frac{6}{5} \right)}}$$

Simplify

The second derivative [src]

       ___ /          1       \ 
-625*\/ 2 *|2 + --------------| 
           \    log(6/5 + 5*x)/ 
--------------------------------
            2   ________________
4*(6 + 25*x) *\/ log(6/5 + 5*x)

$$- \frac{625 \sqrt{2} \left(2 + \frac{1}{\log{\left(5 x + \frac{6}{5} \right)}}\right)}{4 \left(25 x + 6\right)^{2} \sqrt{\log{\left(5 x + \frac{6}{5} \right)}}}$$

Simplify

The third derivative [src]

        ___ /           3                   3        \
15625*\/ 2 *|1 + ---------------- + -----------------|
            |    4*log(6/5 + 5*x)        2           |
            \                       8*log (6/5 + 5*x)/
------------------------------------------------------
                      3   ________________            
            (6 + 25*x) *\/ log(6/5 + 5*x)

$$\frac{15625 \sqrt{2} \left(1 + \frac{3}{4 \log{\left(5 x + \frac{6}{5} \right)}} + \frac{3}{8 \log{\left(5 x + \frac{6}{5} \right)}^{2}}\right)}{\left(25 x + 6\right)^{3} \sqrt{\log{\left(5 x + \frac{6}{5} \right)}}}$$

Simplify

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Derivative of sqrt(2log(5x+1,2))

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The solution