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Identical expressions
(3x^ two + two)(ln^2x)
(3x squared plus 2)(ln squared x)
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3x2+2ln2x
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(3x to the power of 2+2)(ln to the power of 2x)
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Similar expressions
(3x^2-2)(ln^2x)

The derivative of the function/ (3x^2+2)(ln^2x)

Derivative of (3x^2+2)(ln^2x)

The solution

You have entered [src]

/   2    \    2   
\3*x  + 2/*log (x)

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

d //   2    \    2   \
--\\3*x  + 2/*log (x)/
dx

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

Transform

Detail solution

Apply the product rule:

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

$f{\left(x \right)} = 3 x^{2} + 2$ ; to find $\frac{d}{d x} f{\left(x \right)}$ :
1. Differentiate $3 x^{2} + 2$ 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: $x^{2}$ goes to $2 x$
    So, the result is: $6 x$
  2. The derivative of the constant $2$ is zero.
  The result is: $6 x$
$g{\left(x \right)} = \log{\left(x \right)}^{2}$ ; 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: $6 x \log{\left(x \right)}^{2} + \frac{2 \cdot \left(3 x^{2} + 2\right) \log{\left(x \right)}}{x}$
Now simplify:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

  /                                      /       2\\
  |     2                  (-1 + log(x))*\2 + 3*x /|
2*|3*log (x) + 12*log(x) - ------------------------|
  |                                    2           |
  \                                   x            /

$$2 \cdot \left(3 \log{\left(x \right)}^{2} + 12 \log{\left(x \right)} - \frac{\left(3 x^{2} + 2\right) \left(\log{\left(x \right)} - 1\right)}{x^{2}}\right)$$

Simplify

The third derivative [src]

  /                     /       2\\
  |     (-3 + 2*log(x))*\2 + 3*x /|
2*|18 + --------------------------|
  |                  2            |
  \                 x             /
-----------------------------------
                 x

$$\frac{2 \cdot \left(18 + \frac{\left(3 x^{2} + 2\right) \left(2 \log{\left(x \right)} - 3\right)}{x^{2}}\right)}{x}$$

Simplify

The graph

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Derivative of (3x^2+2)(ln^2x)

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Piecewise:

The solution