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Derivative of:
Derivative of x^-6
Derivative of (x+3)/(x-3)
Derivative of (x^3)'
Derivative of log(x^4+x)
Identical expressions
ln((two x+ one)^(one /2))
ln((2x plus 1) to the power of (1 divide by 2))
ln((two x plus one) to the power of (one divide by 2))
ln((2x+1)(1/2))
ln2x+11/2
ln2x+1^1/2
ln((2x+1)^(1 divide by 2))
Similar expressions
ln((2x-1)^(1/2))

The derivative of the function/ ln((2x+1)^(1/2))

Derivative of ln((2x+1)^(1/2))

The solution

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   /  _________\
log\\/ 2*x + 1 /

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

log(sqrt(2*x + 1))

Detail solution

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

$\frac{1}{2 x + 1}$
Now simplify:

$\frac{1}{2 x + 1}$

The answer is:

$\frac{1}{2 x + 1}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

   1   
-------
2*x + 1

$$\frac{1}{2 x + 1}$$

Simplify

The second derivative [src]

   -2     
----------
         2
(1 + 2*x)

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

Simplify

The third derivative [src]

    8     
----------
         3
(1 + 2*x)

$$\frac{8}{\left(2 x + 1\right)^{3}}$$

Simplify