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The derivative of the function/ y=(x²+x+1)²

Derivative of y=(x²+x+1)²

The solution

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            2
/ 2        \ 
\x  + x + 1/

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

(x^2 + x + 1)^2

Detail solution

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

$\left(2 x + 1\right) \left(2 \left(x^{2} + x\right) + 2\right)$
Now simplify:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

          / 2        \
(2 + 4*x)*\x  + x + 1/

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

Simplify

The second derivative [src]

  /             2              \
2*\2 + (1 + 2*x)  + 2*x*(1 + x)/

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

Simplify

The third derivative [src]

12*(1 + 2*x)

$$12 \left(2 x + 1\right)$$

Simplify

The graph

Derivative of y=(x²+x+1)²