Derivative of y=e^x*ln^2x

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 x    2   
e *log (x)

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

d / x    2   \
--\e *log (x)/
dx

$$\frac{d}{d x} e^{x} \log{\left(x \right)}^{2}$$

Transform

Detail solution

Apply the product rule:

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

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

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

The answer is:

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

The graph

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The first derivative [src]

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

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

Simplify

The second derivative [src]

/   2      2*(-1 + log(x))   4*log(x)\  x
|log (x) - --------------- + --------|*e 
|                  2            x    |   
\                 x                  /

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

Simplify

The third derivative [src]

/   2      6*(-1 + log(x))   2*(-3 + 2*log(x))   6*log(x)\  x
|log (x) - --------------- + ----------------- + --------|*e 
|                  2                  3             x    |   
\                 x                  x                   /

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

Simplify

The graph

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Derivative of y=e^x*ln^2x

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The solution