The question asks to find the derivative (도함수) of the function:

$y = \sqrt{x^4 - x^2 + 1}.$

Step-by-Step Solution:

1. Rewrite the square root function: $y = (x^4 - x^2 + 1)^{\frac{1}{2}}.$

2. Apply the chain rule: The derivative of $y$ is: $\frac{dy}{dx} = \frac{1}{2} (x^4 - x^2 + 1)^{-\frac{1}{2}} \cdot \frac{d}{dx}(x^4 - x^2 + 1).$

3. Differentiate the inside term: $\frac{d}{dx}(x^4 - x^2 + 1) = 4x^3 - 2x.$

4. Combine results: Substitute the derivative of the inside term back into the chain rule: $\frac{dy}{dx} = \frac{1}{2} (x^4 - x^2 + 1)^{-\frac{1}{2}} \cdot (4x^3 - 2x).$

5. Simplify the expression: Rewrite the solution as: $\frac{dy}{dx} = \frac{4x^3 - 2x}{2\sqrt{x^4 - x^2 + 1}}.$

Final Answer:

$\frac{dy}{dx} = \frac{4x^3 - 2x}{2\sqrt{x^4 - x^2 + 1}}.$

Let me know if you'd like further details!

Expansion Questions:

1. What is the derivative of $\sqrt{f(x)}$ for general functions $f(x)$?
2. How do you apply the chain rule in a composite function?
3. Can the derivative be further simplified for specific values of $x$?
4. What happens if $x = 0$ in the derivative?
5. How does the result change if the original function includes a constant multiplier?

Tip:

Always identify inner and outer functions when applying the chain rule—it helps avoid errors in differentiation.