Derivative of sqrtxln^2x

You have entered [src]

  ___    2   
\/ x *log (x)

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

sqrt(x)*log(x)^2

Detail solution

Apply the product rule:

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

$f{\left(x \right)} = \sqrt{x}$ ; to find $\frac{d}{d x} f{\left(x \right)}$ :
1. Apply the power rule: $\sqrt{x}$ goes to $\frac{1}{2 \sqrt{x}}$
$g{\left(x \right)} = \log{\left(x \right)}^{2}$ ; to find $\frac{d}{d x} g{\left(x \right)}$ :
1. Let $u = \log{\left(x \right)}$ .
2. Apply the power rule: $u^{2}$ goes to $2 u$
3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \log{\left(x \right)}$ :
  1. The derivative of $\log{\left(x \right)}$ is $\frac{1}{x}$ .
  The result of the chain rule is:
  
  $\frac{2 \log{\left(x \right)}}{x}$
The result is: $\frac{\log{\left(x \right)}^{2}}{2 \sqrt{x}} + \frac{2 \log{\left(x \right)}}{\sqrt{x}}$
Now simplify:

$\frac{\left(\log{\left(x \right)} + 4\right) \log{\left(x \right)}}{2 \sqrt{x}}$

The answer is:

$\frac{\left(\log{\left(x \right)} + 4\right) \log{\left(x \right)}}{2 \sqrt{x}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

   2              
log (x)   2*log(x)
------- + --------
    ___      ___  
2*\/ x     \/ x

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

Simplify

The second derivative [src]

       2   
    log (x)
2 - -------
       4   
-----------
     3/2   
    x

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

Simplify

The third derivative [src]

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

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

Simplify

Other calculators

How to use it?

Identical expressions

Derivative of sqrtxln^2x

The graph:

Piecewise:

The solution