Derivative of (2x²+1)³*ln(1+2x³)

The solution

You have entered [src]

          3              
/   2    \     /       3\
\2*x  + 1/ *log\1 + 2*x /

$$\left(2 x^{2} + 1\right)^{3} \log{\left(2 x^{3} + 1 \right)}$$

(2*x^2 + 1)^3*log(1 + 2*x^3)

Detail solution

Apply the product rule:

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

$f{\left(x \right)} = \left(2 x^{2} + 1\right)^{3}$ ; to find $\frac{d}{d x} f{\left(x \right)}$ :
1. Let $u = 2 x^{2} + 1$ .
2. Apply the power rule: $u^{3}$ goes to $3 u^{2}$
3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(2 x^{2} + 1\right)$ :
  1. Differentiate $2 x^{2} + 1$ 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: $4 x$
    2. The derivative of the constant $1$ is zero.
    The result is: $4 x$
  The result of the chain rule is:
  
  $12 x \left(2 x^{2} + 1\right)^{2}$
$g{\left(x \right)} = \log{\left(2 x^{3} + 1 \right)}$ ; to find $\frac{d}{d x} g{\left(x \right)}$ :
1. Let $u = 2 x^{3} + 1$ .
2. The derivative of $\log{\left(u \right)}$ is $\frac{1}{u}$ .
3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(2 x^{3} + 1\right)$ :
  1. Differentiate $2 x^{3} + 1$ term by term:
    1. The derivative of the constant $1$ is zero.
    2. The derivative of a constant times a function is the constant times the derivative of the function.
      1. Apply the power rule: $x^{3}$ goes to $3 x^{2}$
      So, the result is: $6 x^{2}$
    The result is: $6 x^{2}$
  The result of the chain rule is:
  
  $\frac{6 x^{2}}{2 x^{3} + 1}$
The result is: $\frac{6 x^{2} \left(2 x^{2} + 1\right)^{3}}{2 x^{3} + 1} + 12 x \left(2 x^{2} + 1\right)^{2} \log{\left(2 x^{3} + 1 \right)}$
Now simplify:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

              /                                                           2 /          3  \\
              |                                                 /       2\  |       3*x   ||
              |                                               x*\1 + 2*x / *|-1 + --------||
              |                                3 /       2\                 |            3||
   /       2\ |/        2\    /       3\   12*x *\1 + 2*x /                 \     1 + 2*x /|
12*\1 + 2*x /*|\1 + 10*x /*log\1 + 2*x / + ---------------- - -----------------------------|
              |                                       3                         3          |
              \                                1 + 2*x                   1 + 2*x           /

$$12 \left(2 x^{2} + 1\right) \left(\frac{12 x^{3} \left(2 x^{2} + 1\right)}{2 x^{3} + 1} - \frac{x \left(2 x^{2} + 1\right)^{2} \left(\frac{3 x^{3}}{2 x^{3} + 1} - 1\right)}{2 x^{3} + 1} + \left(10 x^{2} + 1\right) \log{\left(2 x^{3} + 1 \right)}\right)$$

Simplify

The third derivative [src]

   /          3 /         3            6   \                                                                                                   \
   |/       2\  |     18*x         36*x    |                                                   2 /          3  \                               |
   |\1 + 2*x / *|1 - -------- + -----------|                                       2 /       2\  |       3*x   |                               |
   |            |           3             2|                                   36*x *\1 + 2*x / *|-1 + --------|                               |
   |            |    1 + 2*x    /       3\ |                                                     |            3|       2 /       2\ /        2\|
   |            \               \1 + 2*x / /       /        2\    /       3\                     \     1 + 2*x /   18*x *\1 + 2*x /*\1 + 10*x /|
12*|---------------------------------------- + 8*x*\3 + 10*x /*log\1 + 2*x / - --------------------------------- + ----------------------------|
   |                       3                                                                       3                                3          |
   \                1 + 2*x                                                                 1 + 2*x                          1 + 2*x           /

$$12 \left(- \frac{36 x^{2} \left(2 x^{2} + 1\right)^{2} \left(\frac{3 x^{3}}{2 x^{3} + 1} - 1\right)}{2 x^{3} + 1} + \frac{18 x^{2} \left(2 x^{2} + 1\right) \left(10 x^{2} + 1\right)}{2 x^{3} + 1} + 8 x \left(10 x^{2} + 3\right) \log{\left(2 x^{3} + 1 \right)} + \frac{\left(2 x^{2} + 1\right)^{3} \left(\frac{36 x^{6}}{\left(2 x^{3} + 1\right)^{2}} - \frac{18 x^{3}}{2 x^{3} + 1} + 1\right)}{2 x^{3} + 1}\right)$$

Simplify

Other calculators

How to use it?

Identical expressions

Similar expressions

Derivative of (2x²+1)³*ln(1+2x³)

The graph:

Piecewise:

The solution