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Derivative of:
Derivative of x/(e^x+1)
Derivative of (x+1)^1/3
Derivative of sin(4*x-1)^(3)
Derivative of sin(9x+2)
Identical expressions
8sqrt(5x+ one /5x- one)
8 square root of (5x plus 1 divide by 5x minus 1)
8 square root of (5x plus one divide by 5x minus one)
8√(5x+1/5x-1)
8sqrt5x+1/5x-1
8sqrt(5x+1 divide by 5x-1)
Similar expressions
8sqrt(5x-1/5x-1)
8sqrt(5x+1/5x+1)

The derivative of the function/ 8sqrt(5x+1/5x-1)

Derivative of 8sqrt(5x+1/5x-1)

The solution

You have entered [src]

      _____________
     /       x     
8*  /  5*x + - - 1 
  \/         5

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

8*sqrt(5*x + x/5 - 1)

Detail solution

The derivative of a constant times a function is the constant times the derivative of the function.
1. Let $u = \left(\frac{x}{5} + 5 x\right) - 1$ .
2. Apply the power rule: $\sqrt{u}$ goes to $\frac{1}{2 \sqrt{u}}$
3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(\left(\frac{x}{5} + 5 x\right) - 1\right)$ :
  1. Differentiate $\left(\frac{x}{5} + 5 x\right) - 1$ term by term:
    1. Differentiate $\frac{x}{5} + 5 x$ 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 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: $\frac{1}{5}$
      The result is: $\frac{26}{5}$
    2. The derivative of the constant $-1$ is zero.
    The result is: $\frac{26}{5}$
  The result of the chain rule is:
  
  $\frac{13}{5 \sqrt{\left(\frac{x}{5} + 5 x\right) - 1}}$
So, the result is: $\frac{104}{5 \sqrt{\left(\frac{x}{5} + 5 x\right) - 1}}$
Now simplify:

$\frac{104 \sqrt{5}}{5 \sqrt{26 x - 5}}$

The answer is:

$\frac{104 \sqrt{5}}{5 \sqrt{26 x - 5}}$

The graph

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The first derivative [src]

        104        
-------------------
      _____________
     /       x     
5*  /  5*x + - - 1 
  \/         5

$$\frac{104}{5 \sqrt{\left(\frac{x}{5} + 5 x\right) - 1}}$$

Simplify

The second derivative [src]

      -1352      
-----------------
              3/2
   /     26*x\   
25*|-1 + ----|   
   \      5  /

$$- \frac{1352}{25 \left(\frac{26 x}{5} - 1\right)^{\frac{3}{2}}}$$

Simplify

The third derivative [src]

      52728       
------------------
               5/2
    /     26*x\   
125*|-1 + ----|   
    \      5  /

$$\frac{52728}{125 \left(\frac{26 x}{5} - 1\right)^{\frac{5}{2}}}$$

Simplify