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Derivative of:
Derivative of x^3+1/x-1
Derivative of x^2/cosx
Derivative of x^2/2x+1
Derivative of sin(x)+cot(x)
Identical expressions
(sqrt(one +x)*ln(one +x))
( square root of (1 plus x) multiply by ln(1 plus x))
( square root of (one plus x) multiply by ln(one plus x))
(√(1+x)*ln(1+x))
(sqrt(1+x)ln(1+x))
sqrt1+xln1+x
Similar expressions
(sqrt(1+x)*ln(1-x))
(sqrt(1-x)*ln(1+x))

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

Derivative of (sqrt(1+x)*ln(1+x))

The solution

You have entered [src]

  _______           
\/ 1 + x *log(1 + x)

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

sqrt(1 + x)*log(1 + x)

Detail solution

Apply the product rule:

$\frac{d}{d x} f{\left(x \right)} g{\left(x \right)} = f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}$

$f{\left(x \right)} = \sqrt{x + 1}$ ; to find $\frac{d}{d x} f{\left(x \right)}$ :
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:
  
  $\frac{1}{2 \sqrt{x + 1}}$
$g{\left(x \right)} = \log{\left(x + 1 \right)}$ ; to find $\frac{d}{d x} g{\left(x \right)}$ :
1. Let $u = x + 1$ .
2. The derivative of $\log{\left(u \right)}$ is $\frac{1}{u}$ .
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:
  
  $\frac{1}{x + 1}$
The result is: $\frac{\log{\left(x + 1 \right)}}{2 \sqrt{x + 1}} + \frac{1}{\sqrt{x + 1}}$
Now simplify:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

-2 + 3*log(1 + x)
-----------------
            5/2  
   8*(1 + x)

$$\frac{3 \log{\left(x + 1 \right)} - 2}{8 \left(x + 1\right)^{\frac{5}{2}}}$$

Simplify

3-я производная [src]

-2 + 3*log(1 + x)
-----------------
            5/2  
   8*(1 + x)

$$\frac{3 \log{\left(x + 1 \right)} - 2}{8 \left(x + 1\right)^{\frac{5}{2}}}$$

Simplify