Certainly! Let's walk through the steps to differentiate [tex]\( f(x) = \sqrt{x^4 - 6x^2 + 8} \)[/tex].
To find the derivative [tex]\( f'(x) \)[/tex]:
1. Rewrite the Function:
Rewrite [tex]\( f(x) \)[/tex] as [tex]\( (x^4 - 6x^2 + 8)^{1/2} \)[/tex].
2. Apply the Chain Rule:
We'll use the chain rule: if [tex]\( y = g(h(x)) \)[/tex], then [tex]\( y' = g'(h(x)) \cdot h'(x) \)[/tex].
Here, [tex]\( g(u) = u^{1/2} \)[/tex] where [tex]\( u = x^4 - 6x^2 + 8 \)[/tex].
3. Differentiate [tex]\( g(u) \)[/tex]:
[tex]\( g(u) = \sqrt{u} = u^{1/2} \)[/tex].
Hence, [tex]\( g'(u) = \frac{1}{2} u^{-1/2} = \frac{1}{2 \sqrt{u}} \)[/tex].
4. Differentiate [tex]\( u \)[/tex] with respect to [tex]\( x \)[/tex]:
[tex]\( u = x^4 - 6x^2 + 8 \)[/tex]
Differentiating [tex]\( u \)[/tex] with respect to [tex]\( x \)[/tex] gives [tex]\( u' = 4x^3 - 12x \)[/tex].
5. Combine Using the Chain Rule:
Applying the chain rule, we get:
[tex]\[
f'(x) = g'(u) \cdot u' = \frac{1}{2 \sqrt{u}} \cdot (4x^3 - 12x)
\][/tex]
6. Substitute Back [tex]\( u = x^4 - 6x^2 + 8 \)[/tex]:
[tex]\[
f'(x) = \frac{1}{2 \sqrt{x^4 - 6x^2 + 8}} \cdot (4x^3 - 12x)
\][/tex]
7. Simplify the Expression:
Factor out common terms in the numerator:
[tex]\[
f'(x) = \frac{1}{2 \sqrt{x^4 - 6x^2 + 8}} \cdot 4x(x^2 - 3)
\][/tex]
[tex]\[
f'(x) = \frac{4x(x^2 - 3)}{2 \sqrt{x^4 - 6x^2 + 8}}
\][/tex]
Simplify further:
[tex]\[
f'(x) = \frac{2x(x^2 - 3)}{\sqrt{x^4 - 6x^2 + 8}}
\][/tex]
Therefore, the derivative of the function is:
[tex]\[
f'(x) = \frac{2x(x^2 - 3)}{\sqrt{x^4 - 6x^2 + 8}}
\][/tex]
This is the simplified form of the derivative.
