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1/(3*sqrt(x+1))

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Identical expressions
one /(three *sqrt(x+ one))
1 divide by (3 multiply by square root of (x plus 1))
one divide by (three multiply by square root of (x plus one))
1/(3*√(x+1))
1/(3sqrt(x+1))
1/3sqrtx+1
1 divide by (3*sqrt(x+1))
Similar expressions
1/(3*sqrt(x-1))

The derivative of the function/ 1/(3*sqrt(x+1))

Derivative of 1/(3*sqrt(x+1))

The solution

You have entered [src]

     1     
-----------
    _______
3*\/ x + 1

$$\frac{1}{3 \sqrt{x + 1}}$$

1/(3*sqrt(x + 1))

Detail solution

Let $u = 3 \sqrt{x + 1}$ .
Apply the power rule: $\frac{1}{u}$ goes to $- \frac{1}{u^{2}}$
Then, apply the chain rule. Multiply by $\frac{d}{d x} 3 \sqrt{x + 1}$ :
1. The derivative of a constant times a function is the constant times the derivative of the function.
  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. Apply the power rule: $x$ goes to $1$
      2. The derivative of the constant $1$ is zero.
      The result is: $1$
    The result of the chain rule is:
    
    $\frac{1}{2 \sqrt{x + 1}}$
  So, the result is: $\frac{3}{2 \sqrt{x + 1}}$
The result of the chain rule is:

$- \frac{1}{6 \left(x + 1\right)^{\frac{3}{2}}}$
Now simplify:

$- \frac{1}{6 \left(x + 1\right)^{\frac{3}{2}}}$

The answer is:

$- \frac{1}{6 \left(x + 1\right)^{\frac{3}{2}}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

    -5      
------------
         7/2
8*(1 + x)

$$- \frac{5}{8 \left(x + 1\right)^{\frac{7}{2}}}$$

Simplify

The graph

Derivative of 1/(3*sqrt(x+1))