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

You have entered [src]

  ________________
\/ log(5*x + 6/5)

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

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

Detail solution

Let $u = \log{\left(5 x + \frac{6}{5} \right)}$ .
Apply the power rule: $\sqrt{u}$ goes to $\frac{1}{2 \sqrt{u}}$
Then, apply the chain rule. Multiply by $\frac{d}{d x} \log{\left(5 x + \frac{6}{5} \right)}$ :
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.
      1. 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:
  
  $\frac{5}{5 x + \frac{6}{5}}$
The result of the chain rule is:

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

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

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

Simplify

Other calculators

How to use it?

Identical expressions

Similar expressions

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

The graph:

Piecewise:

The solution