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Identical expressions
(6x+ two)^ two *ln(2x+ seven)
(6x plus 2) squared multiply by ln(2x plus 7)
(6x plus two) to the power of two multiply by ln(2x plus seven)
(6x+2)2*ln(2x+7)
6x+22*ln2x+7
(6x+2)²*ln(2x+7)
(6x+2) to the power of 2*ln(2x+7)
(6x+2)^2ln(2x+7)
(6x+2)2ln(2x+7)
6x+22ln2x+7
6x+2^2ln2x+7
Similar expressions
(6x-2)^2*ln(2x+7)
(6x+2)^2*ln(2x-7)

The derivative of the function/ (6x+2)^2*ln(2x+7)

Derivative of (6x+2)^2*ln(2x+7)

The solution

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         2             
(6*x + 2) *log(2*x + 7)

$$\left(6 x + 2\right)^{2} \log{\left(2 x + 7 \right)}$$

(6*x + 2)^2*log(2*x + 7)

Detail solution

Apply the product rule:

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

$f{\left(x \right)} = \left(6 x + 2\right)^{2}$ ; to find $\frac{d}{d x} f{\left(x \right)}$ :
1. Let $u = 6 x + 2$ .
2. Apply the power rule: $u^{2}$ goes to $2 u$
3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(6 x + 2\right)$ :
  1. Differentiate $6 x + 2$ 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: $x$ goes to $1$
      So, the result is: $6$
    2. The derivative of the constant $2$ is zero.
    The result is: $6$
  The result of the chain rule is:
  
  $72 x + 24$
$g{\left(x \right)} = \log{\left(2 x + 7 \right)}$ ; to find $\frac{d}{d x} g{\left(x \right)}$ :
1. Let $u = 2 x + 7$ .
2. The derivative of $\log{\left(u \right)}$ is $\frac{1}{u}$ .
3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(2 x + 7\right)$ :
  1. Differentiate $2 x + 7$ 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: $x$ goes to $1$
      So, the result is: $2$
    2. The derivative of the constant $7$ is zero.
    The result is: $2$
  The result of the chain rule is:
  
  $\frac{2}{2 x + 7}$
The result is: $\left(72 x + 24\right) \log{\left(2 x + 7 \right)} + \frac{2 \left(6 x + 2\right)^{2}}{2 x + 7}$
Now simplify:

$\frac{8 \left(3 x + 1\right) \left(3 x + 3 \left(2 x + 7\right) \log{\left(2 x + 7 \right)} + 1\right)}{2 x + 7}$

The answer is:

$\frac{8 \left(3 x + 1\right) \left(3 x + 3 \left(2 x + 7\right) \log{\left(2 x + 7 \right)} + 1\right)}{2 x + 7}$

The graph

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The first derivative [src]

                                      2
                           2*(6*x + 2) 
(24 + 72*x)*log(2*x + 7) + ------------
                             2*x + 7

$$\left(72 x + 24\right) \log{\left(2 x + 7 \right)} + \frac{2 \left(6 x + 2\right)^{2}}{2 x + 7}$$

Simplify

The second derivative [src]

  /                            2               \
  |                 2*(1 + 3*x)    12*(1 + 3*x)|
8*|9*log(7 + 2*x) - ------------ + ------------|
  |                           2      7 + 2*x   |
  \                  (7 + 2*x)                 /

$$8 \left(9 \log{\left(2 x + 7 \right)} + \frac{12 \left(3 x + 1\right)}{2 x + 7} - \frac{2 \left(3 x + 1\right)^{2}}{\left(2 x + 7\right)^{2}}\right)$$

Simplify

The third derivative [src]

   /                               2\
   |     18*(1 + 3*x)   4*(1 + 3*x) |
16*|27 - ------------ + ------------|
   |       7 + 2*x                2 |
   \                     (7 + 2*x)  /
-------------------------------------
               7 + 2*x

$$\frac{16 \left(27 - \frac{18 \left(3 x + 1\right)}{2 x + 7} + \frac{4 \left(3 x + 1\right)^{2}}{\left(2 x + 7\right)^{2}}\right)}{2 x + 7}$$

Simplify

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Derivative of (6x+2)^2*ln(2x+7)

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Piecewise:

The solution