Derivative of y=(√x-1)²-(x²+1)⁴

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           2           4
/  ___    \    / 2    \ 
\\/ x  - 1/  - \x  + 1/

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

  /           2           4\
d |/  ___    \    / 2    \ |
--\\\/ x  - 1/  - \x  + 1/ /
dx

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

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Detail solution

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

$- 8 x \left(x^{2} + 1\right)^{3} + 1 - \frac{1}{\sqrt{x}}$

The answer is:

$- 8 x \left(x^{2} + 1\right)^{3} + 1 - \frac{1}{\sqrt{x}}$

The graph

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The first derivative [src]

  ___                   3
\/ x  - 1       / 2    \ 
--------- - 8*x*\x  + 1/ 
    ___                  
  \/ x

$$- 8 x \left(x^{2} + 1\right)^{3} + \frac{\sqrt{x} - 1}{\sqrt{x}}$$

Simplify

The second derivative [src]

                3                 2          ___
 1      /     2\        2 /     2\    -1 + \/ x 
--- - 8*\1 + x /  - 48*x *\1 + x /  - ----------
2*x                                        3/2  
                                        2*x

$$- 48 x^{2} \left(x^{2} + 1\right)^{2} - 8 \left(x^{2} + 1\right)^{3} + \frac{1}{2 x} - \frac{\sqrt{x} - 1}{2 x^{\frac{3}{2}}}$$

Simplify

The third derivative [src]

  /                                       2          ___\
  |   1         3 /     2\        /     2\    -1 + \/ x |
3*|- ---- - 64*x *\1 + x / - 48*x*\1 + x /  + ----------|
  |     2                                          5/2  |
  \  4*x                                        4*x     /

$$3 \left(- 64 x^{3} \left(x^{2} + 1\right) - 48 x \left(x^{2} + 1\right)^{2} - \frac{1}{4 x^{2}} + \frac{\sqrt{x} - 1}{4 x^{\frac{5}{2}}}\right)$$

Simplify

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Derivative of y=(√x-1)²-(x²+1)⁴

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