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Similar expressions
(log^3)(2x-3)^2

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

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

The solution

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

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

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

Detail solution

Apply the product rule:

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

$f{\left(x \right)} = \log{\left(x \right)}^{3}$ ; to find $\frac{d}{d x} f{\left(x \right)}$ :
1. Let $u = \log{\left(x \right)}$ .
2. Apply the power rule: $u^{3}$ goes to $3 u^{2}$
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:
  
  $\frac{3 \log{\left(x \right)}^{2}}{x}$
$g{\left(x \right)} = \left(2 x + 3\right)^{2}$ ; to find $\frac{d}{d x} g{\left(x \right)}$ :
1. Let $u = 2 x + 3$ .
2. Apply the power rule: $u^{2}$ goes to $2 u$
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.
      1. 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:
  
  $8 x + 12$
The result is: $\left(8 x + 12\right) \log{\left(x \right)}^{3} + \frac{3 \left(2 x + 3\right)^{2} \log{\left(x \right)}^{2}}{x}$
Now simplify:

$\frac{\left(2 x + 3\right) \left(4 x \log{\left(x \right)} + 6 x + 9\right) \log{\left(x \right)}^{2}}{x}$

The answer is:

$\frac{\left(2 x + 3\right) \left(4 x \log{\left(x \right)} + 6 x + 9\right) \log{\left(x \right)}^{2}}{x}$

The graph

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Plot the graph f'(x)

The first derivative [src]

                                2    2   
   3                 3*(2*x + 3) *log (x)
log (x)*(12 + 8*x) + --------------------
                              x

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

Simplify

The second derivative [src]

/                       2                                    \       
|     2      3*(3 + 2*x) *(-2 + log(x))   24*(3 + 2*x)*log(x)|       
|8*log (x) - -------------------------- + -------------------|*log(x)
|                         2                        x         |       
\                        x                                   /

$$\left(8 \log{\left(x \right)}^{2} + \frac{24 \left(2 x + 3\right) \log{\left(x \right)}}{x} - \frac{3 \left(2 x + 3\right)^{2} \left(\log{\left(x \right)} - 2\right)}{x^{2}}\right) \log{\left(x \right)}$$

Simplify

The third derivative [src]

  /                      2 /       2              \                                   \
  |      2      (3 + 2*x) *\1 + log (x) - 3*log(x)/   6*(-2 + log(x))*(3 + 2*x)*log(x)|
6*|12*log (x) + ----------------------------------- - --------------------------------|
  |                               2                                  x                |
  \                              x                                                    /
---------------------------------------------------------------------------------------
                                           x

$$\frac{6 \left(12 \log{\left(x \right)}^{2} - \frac{6 \left(2 x + 3\right) \left(\log{\left(x \right)} - 2\right) \log{\left(x \right)}}{x} + \frac{\left(2 x + 3\right)^{2} \left(\log{\left(x \right)}^{2} - 3 \log{\left(x \right)} + 1\right)}{x^{2}}\right)}{x}$$

Simplify

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

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The solution