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

Function f() - derivative -N order at the point
v

The graph:

from to

Piecewise:

The solution

You have entered [src]
   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
  1. Apply the product rule:

    ; to find :

    1. Let .

    2. Apply the power rule: goes to

    3. Then, apply the chain rule. Multiply by :

      1. The derivative of is .

      The result of the chain rule is:

    ; to find :

    1. Let .

    2. Apply the power rule: goes to

    3. Then, apply the chain rule. Multiply by :

      1. Differentiate 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: goes to

          So, the result is:

        2. The derivative of the constant is zero.

        The result is:

      The result of the chain rule is:

    The result is:

  2. Now simplify:


The answer is:

The graph
The first derivative [src]
                                2    2   
   3                 3*(2*x + 3) *log (x)
log (x)*(12 + 8*x) + --------------------
                              x          
$$\left(8 x + 12\right) \log{\left(x \right)}^{3} + \frac{3 \left(2 x + 3\right)^{2} \log{\left(x \right)}^{2}}{x}$$
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                                   /       
$$\left(8 \log{\left(x \right)}^{2} + \frac{24 \left(2 x + 3\right) \log{\left(x \right)}}{x} - \frac{3 \left(2 x + 3\right)^{2} \left(\log{\left(x \right)} - 2\right)}{x^{2}}\right) \log{\left(x \right)}$$
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                                           
$$\frac{6 \left(12 \log{\left(x \right)}^{2} - \frac{6 \left(2 x + 3\right) \left(\log{\left(x \right)} - 2\right) \log{\left(x \right)}}{x} + \frac{\left(2 x + 3\right)^{2} \left(\log{\left(x \right)}^{2} - 3 \log{\left(x \right)} + 1\right)}{x^{2}}\right)}{x}$$