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

Derivative of 1/sqrt(1+x)

The solution

You have entered [src]

    1    
---------
  _______
\/ 1 + x

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

1/(sqrt(1 + x))

Detail solution

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

        -1         
-------------------
            _______
2*(1 + x)*\/ 1 + x

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

Simplify

The second derivative [src]

     3      
------------
         5/2
4*(1 + x)

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

Simplify

The third derivative [src]

    -15     
------------
         7/2
8*(1 + x)

$$- \frac{15}{8 \left(x + 1\right)^{\frac{7}{2}}}$$

Simplify

The graph

Derivative of 1/sqrt(1+x)