Derivative of y=(x²+4)²(2x³-1)³

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        2           3
/ 2    \  /   3    \ 
\x  + 4/ *\2*x  - 1/

$$\left(x^{2} + 4\right)^{2} \left(2 x^{3} - 1\right)^{3}$$

  /        2           3\
d |/ 2    \  /   3    \ |
--\\x  + 4/ *\2*x  - 1/ /
dx

$$\frac{d}{d x} \left(x^{2} + 4\right)^{2} \left(2 x^{3} - 1\right)^{3}$$

Transform

Detail solution

Apply the product rule:

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

$f{\left(x \right)} = \left(x^{2} + 4\right)^{2}$ ; to find $\frac{d}{d x} f{\left(x \right)}$ :
1. Let $u = x^{2} + 4$ .
2. Apply the power rule: $u^{2}$ goes to $2 u$
3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(x^{2} + 4\right)$ :
  1. Differentiate $x^{2} + 4$ term by term:
    1. Apply the power rule: $x^{2}$ goes to $2 x$
    2. The derivative of the constant $4$ is zero.
    The result is: $2 x$
  The result of the chain rule is:
  
  $2 x \left(2 x^{2} + 8\right)$
$g{\left(x \right)} = \left(2 x^{3} - 1\right)^{3}$ ; to find $\frac{d}{d x} g{\left(x \right)}$ :
1. Let $u = 2 x^{3} - 1$ .
2. Apply the power rule: $u^{3}$ goes to $3 u^{2}$
3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(2 x^{3} - 1\right)$ :
  1. Differentiate $2 x^{3} - 1$ 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: $x^{3}$ goes to $3 x^{2}$
      So, the result is: $6 x^{2}$
    2. The derivative of the constant $\left(-1\right) 1$ is zero.
    The result is: $6 x^{2}$
  The result of the chain rule is:
  
  $18 x^{2} \left(2 x^{3} - 1\right)^{2}$
The result is: $18 x^{2} \left(x^{2} + 4\right)^{2} \left(2 x^{3} - 1\right)^{2} + 2 x \left(2 x^{2} + 8\right) \left(2 x^{3} - 1\right)^{3}$
Now simplify:

$2 x \left(x^{2} + 4\right) \left(2 x^{3} - 1\right)^{2} \cdot \left(4 x^{3} + 9 x \left(x^{2} + 4\right) - 2\right)$

The answer is:

$2 x \left(x^{2} + 4\right) \left(2 x^{3} - 1\right)^{2} \cdot \left(4 x^{3} + 9 x \left(x^{2} + 4\right) - 2\right)$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

              3                          2           2
    /   3    \  / 2    \       2 / 2    \  /   3    \ 
4*x*\2*x  - 1/ *\x  + 4/ + 18*x *\x  + 4/ *\2*x  - 1/

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

Simplify

The second derivative [src]

              /           2                          2                                         \
  /        3\ |/        3\  /       2\       /     2\  /        3\       3 /        3\ /     2\|
4*\-1 + 2*x /*\\-1 + 2*x / *\4 + 3*x / + 9*x*\4 + x / *\-1 + 8*x / + 36*x *\-1 + 2*x /*\4 + x //

$$4 \cdot \left(2 x^{3} - 1\right) \left(36 x^{3} \left(x^{2} + 4\right) \left(2 x^{3} - 1\right) + 9 x \left(x^{2} + 4\right)^{2} \cdot \left(8 x^{3} - 1\right) + \left(3 x^{2} + 4\right) \left(2 x^{3} - 1\right)^{2}\right)$$

Simplify

The third derivative [src]

   /               3             2 /           2                            \                    2                                                    \
   |    /        3\      /     2\  |/        3\        6       3 /        3\|       2 /        3\  /       2\       2 /        3\ /        3\ /     2\|
12*\2*x*\-1 + 2*x /  + 3*\4 + x / *\\-1 + 2*x /  + 36*x  + 36*x *\-1 + 2*x // + 18*x *\-1 + 2*x / *\4 + 3*x / + 36*x *\-1 + 2*x /*\-1 + 8*x /*\4 + x //

$$12 \cdot \left(36 x^{2} \left(x^{2} + 4\right) \left(2 x^{3} - 1\right) \left(8 x^{3} - 1\right) + 18 x^{2} \cdot \left(3 x^{2} + 4\right) \left(2 x^{3} - 1\right)^{2} + 2 x \left(2 x^{3} - 1\right)^{3} + 3 \left(x^{2} + 4\right)^{2} \cdot \left(36 x^{6} + 36 x^{3} \cdot \left(2 x^{3} - 1\right) + \left(2 x^{3} - 1\right)^{2}\right)\right)$$

Simplify

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Derivative of y=(x²+4)²(2x³-1)³

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The solution