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Similar expressions
(1-ln^2(x))*(ln(x))

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

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

The solution

You have entered [src]

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

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

(1 + log(x)^2)*log(x)

Detail solution

Apply the product rule:

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

$f{\left(x \right)} = \log{\left(x \right)}^{2} + 1$ ; to find $\frac{d}{d x} f{\left(x \right)}$ :
1. Differentiate $\log{\left(x \right)}^{2} + 1$ term by term:
  1. The derivative of the constant $1$ is zero.
  2. Let $u = \log{\left(x \right)}$ .
  3. Apply the power rule: $u^{2}$ goes to $2 u$
  4. 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: $\frac{2 \log{\left(x \right)}}{x}$
$g{\left(x \right)} = \log{\left(x \right)}$ ; to find $\frac{d}{d x} g{\left(x \right)}$ :
1. The derivative of $\log{\left(x \right)}$ is $\frac{1}{x}$ .
The result is: $\frac{\log{\left(x \right)}^{2} + 1}{x} + \frac{2 \log{\left(x \right)}^{2}}{x}$
Now simplify:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

  /       2                                       \
2*\4 + log (x) - 6*log(x) + (-3 + 2*log(x))*log(x)/
---------------------------------------------------
                          3                        
                         x

$$\frac{2 \left(\left(2 \log{\left(x \right)} - 3\right) \log{\left(x \right)} + \log{\left(x \right)}^{2} - 6 \log{\left(x \right)} + 4\right)}{x^{3}}$$

Simplify

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

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The solution