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Identical expressions
(log(two x+ three)^2)^ three
( logarithm of (2x plus 3) squared ) cubed
( logarithm of (two x plus three) squared ) to the power of three
(log(2x+3)2)3
log2x+323
(log(2x+3)²)³
(log(2x+3) to the power of 2) to the power of 3
log2x+3^2^3
Similar expressions
(log(2x-3)^2)^3

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

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

The solution

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             3
   2          
log (2*x + 3)

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

(log(2*x + 3)^2)^3

Detail solution

Let $u = \log{\left(2 x + 3 \right)}^{2}$ .
Apply the power rule: $u^{3}$ goes to $3 u^{2}$
Then, apply the chain rule. Multiply by $\frac{d}{d x} \log{\left(2 x + 3 \right)}^{2}$ :
1. Let $u = \log{\left(2 x + 3 \right)}$ .
2. Apply the power rule: $u^{2}$ goes to $2 u$
3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \log{\left(2 x + 3 \right)}$ :
  1. Let $u = 2 x + 3$ .
  2. The derivative of $\log{\left(u \right)}$ is $\frac{1}{u}$ .
  3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(2 x + 3\right)$ :
    1. Differentiate $2 x + 3$ term by term:
      1. The derivative of a constant times a function is the constant times the derivative of the function.
        
        Apply the power rule: $x$ goes to $1$
        
        So, the result is: $2$
      2. The derivative of the constant $3$ is zero.
      The result is: $2$
    The result of the chain rule is:
    
    $\frac{2}{2 x + 3}$
  The result of the chain rule is:
  
  $\frac{4 \log{\left(2 x + 3 \right)}}{2 x + 3}$
The result of the chain rule is:

$\frac{12 \log{\left(2 x + 3 \right)}^{5}}{2 x + 3}$
Now simplify:

$\frac{12 \log{\left(2 x + 3 \right)}^{5}}{2 x + 3}$

The answer is:

$\frac{12 \log{\left(2 x + 3 \right)}^{5}}{2 x + 3}$

The graph

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The first derivative [src]

         6            
   12*log (2*x + 3)   
----------------------
(2*x + 3)*log(2*x + 3)

$$\frac{12 \log{\left(2 x + 3 \right)}^{6}}{\left(2 x + 3\right) \log{\left(2 x + 3 \right)}}$$

Simplify

The second derivative [src]

      4                            
24*log (3 + 2*x)*(5 - log(3 + 2*x))
-----------------------------------
                      2            
             (3 + 2*x)

$$\frac{24 \left(5 - \log{\left(2 x + 3 \right)}\right) \log{\left(2 x + 3 \right)}^{4}}{\left(2 x + 3\right)^{2}}$$

Simplify

The third derivative [src]

      3          /                            2         \
48*log (3 + 2*x)*\20 - 15*log(3 + 2*x) + 2*log (3 + 2*x)/
---------------------------------------------------------
                                 3                       
                        (3 + 2*x)

$$\frac{48 \left(2 \log{\left(2 x + 3 \right)}^{2} - 15 \log{\left(2 x + 3 \right)} + 20\right) \log{\left(2 x + 3 \right)}^{3}}{\left(2 x + 3\right)^{3}}$$

Simplify

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

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The solution