To find the derivative of the function $y = \frac{x^4}{\sqrt{x^4 - 1}}$, we will apply the quotient rule and chain rule.

Step 1: Identify the components

Let: $u(x) = x^4 \quad \text{and} \quad v(x) = \sqrt{x^4 - 1}$ The derivative of $y$ with respect to $x$ is given by the quotient rule:

$y' = \frac{u'(x)v(x) - u(x)v'(x)}{[v(x)]^2}$

Step 2: Differentiate $u(x)$ and $v(x)$

1. Derivative of $u(x) = x^4$: $u'(x) = 4x^3$

2. Derivative of $v(x) = \sqrt{x^4 - 1}$:

   • First, express $v(x)$ as $v(x) = (x^4 - 1)^{1/2}$.
   • Use the chain rule: $v'(x) = \frac{1}{2}(x^4 - 1)^{-1/2} \cdot 4x^3$.

$v'(x) = \frac{4x^3}{2\sqrt{x^4 - 1}} = \frac{2x^3}{\sqrt{x^4 - 1}}$

Step 3: Apply the Quotient Rule

Substitute $u'(x)$, $u(x)$, $v(x)$, and $v'(x)$ into the quotient rule formula:

$y' = \frac{(4x^3)\sqrt{x^4 - 1} - \left(x^4 \cdot \frac{2x^3}{\sqrt{x^4 - 1}}\right)}{(\sqrt{x^4 - 1})^2}$

Step 4: Simplify the expression

Simplify the numerator and denominator:

• The numerator: $4x^3\sqrt{x^4 - 1} - \frac{2x^7}{\sqrt{x^4 - 1}} = \frac{4x^3(x^4 - 1) - 2x^7}{\sqrt{x^4 - 1}}$ $= \frac{4x^7 - 4x^3 - 2x^7}{\sqrt{x^4 - 1}} = \frac{2x^7 - 4x^3}{\sqrt{x^4 - 1}} = \frac{2x^3(x^4 - 2)}{\sqrt{x^4 - 1}}$

• The denominator: $(\sqrt{x^4 - 1})^2 = x^4 - 1$

So the derivative $y'$ becomes:

$y' = \frac{2x^3(x^4 - 2)}{(x^4 - 1)^{3/2}}$

Final Answer:

$\boxed{y' = \frac{2x^3(x^4 - 2)}{(x^4 - 1)^{3/2}}}$

Do you need any additional details or explanations?

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Relative Questions:

1. How would you find the second derivative of the given function?
2. What is the significance of the chain rule in finding derivatives?
3. How do you interpret the derivative of a function in terms of its graph?
4. Can you simplify the derivative further if given a specific value of $x$?
5. How would you find the critical points of the function $y(x)$?
6. What does the derivative tell us about the concavity of the function?
7. How would you approach finding the integral of this function?
8. What happens to the derivative as $x$ approaches infinity?

Tip: When applying the quotient rule, always simplify the expression step by step to avoid errors in complex derivatives.