Derivative of y=ln(sqrt(5x)-1)

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   /  _____    \
log\\/ 5*x  - 1/

$$\log{\left(\sqrt{5 x} - 1 \right)}$$

log(sqrt(5*x) - 1)

Detail solution

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

$\frac{\sqrt{5}}{2 \sqrt{x} \left(\sqrt{5 x} - 1\right)}$
Now simplify:

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

The answer is:

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

The graph

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The first derivative [src]

          ___        
        \/ 5         
---------------------
    ___ /  _____    \
2*\/ x *\\/ 5*x  - 1/

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

Simplify

The second derivative [src]

 /  ___                       \ 
 |\/ 5             5          | 
-|----- + --------------------| 
 |  3/2     /       ___   ___\| 
 \ x      x*\-1 + \/ 5 *\/ x // 
--------------------------------
        /       ___   ___\      
      4*\-1 + \/ 5 *\/ x /

$$- \frac{\frac{5}{x \left(\sqrt{5} \sqrt{x} - 1\right)} + \frac{\sqrt{5}}{x^{\frac{3}{2}}}}{4 \left(\sqrt{5} \sqrt{x} - 1\right)}$$

Simplify

The third derivative [src]

    ___                                        ___        
3*\/ 5              15                    10*\/ 5         
------- + --------------------- + ------------------------
   5/2     2 /       ___   ___\                          2
  x       x *\-1 + \/ 5 *\/ x /    3/2 /       ___   ___\ 
                                  x   *\-1 + \/ 5 *\/ x / 
----------------------------------------------------------
                     /       ___   ___\                   
                   8*\-1 + \/ 5 *\/ x /

$$\frac{\frac{15}{x^{2} \left(\sqrt{5} \sqrt{x} - 1\right)} + \frac{10 \sqrt{5}}{x^{\frac{3}{2}} \left(\sqrt{5} \sqrt{x} - 1\right)^{2}} + \frac{3 \sqrt{5}}{x^{\frac{5}{2}}}}{8 \left(\sqrt{5} \sqrt{x} - 1\right)}$$

Simplify

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Derivative of y=ln(sqrt(5x)-1)

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