Derivative of sqrt(1/x)

You have entered [src]

    ___
   / 1 
  /  - 
\/   x

$$\sqrt{\frac{1}{x}}$$

sqrt(1/x)

Detail solution

Let $u = \frac{1}{x}$ .
Apply the power rule: $\sqrt{u}$ goes to $\frac{1}{2 \sqrt{u}}$
Then, apply the chain rule. Multiply by $\frac{d}{d x} \frac{1}{x}$ :
1. Apply the power rule: $\frac{1}{x}$ goes to $- \frac{1}{x^{2}}$
The result of the chain rule is:

$- \frac{1}{2 x^{2} \sqrt{\frac{1}{x}}}$

The answer is:

$- \frac{1}{2 x^{2} \sqrt{\frac{1}{x}}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

$$- \frac{\sqrt{\frac{1}{x}}}{2 x}$$

Simplify

The second derivative [src]

      ___
     / 1 
3*  /  - 
  \/   x 
---------
      2  
   4*x

$$\frac{3 \sqrt{\frac{1}{x}}}{4 x^{2}}$$

Simplify

The third derivative [src]

        ___
       / 1 
-15*  /  - 
    \/   x 
-----------
       3   
    8*x

$$- \frac{15 \sqrt{\frac{1}{x}}}{8 x^{3}}$$

Simplify

The graph