Question: Calculate the derivative indicated.
$$\frac{d^2y}{dx^2} \bigg|_{x=6}$$
where $$y = \frac{8}{x^3} + x^2$$
1. **State the problem:** We need to find the second derivative of $$y = \frac{8}{x^3} + x^2$$ with respect to $$x$$ and then evaluate it at $$x=6$$.
2. **Rewrite the function:** Express $$y$$ in terms of powers of $$x$$ for easier differentiation:
$$y = 8x^{-3} + x^2$$
3. **First derivative:** Use the power rule $$\frac{d}{dx} x^n = nx^{n-1}$$:
$$\frac{dy}{dx} = 8 \cdot (-3)x^{-4} + 2x = -24x^{-4} + 2x$$
4. **Second derivative:** Differentiate again:
$$\frac{d^2y}{dx^2} = \frac{d}{dx}(-24x^{-4} + 2x) = -24 \cdot (-4)x^{-5} + 2 = 96x^{-5} + 2$$
5. **Evaluate at $$x=6$$:**
$$\frac{d^2y}{dx^2} \bigg|_{x=6} = 96 \cdot 6^{-5} + 2 = 96 \cdot \frac{1}{6^5} + 2$$
Calculate $$6^5$$:
$$6^5 = 6 \times 6 \times 6 \times 6 \times 6 = 7776$$
So:
$$\frac{d^2y}{dx^2} \bigg|_{x=6} = \frac{96}{7776} + 2$$
Simplify the fraction by dividing numerator and denominator by 24:
$$\frac{\cancel{96}^{4}}{\cancel{7776}^{324}} + 2 = \frac{4}{324} + 2$$
Simplify $$\frac{4}{324}$$ by dividing numerator and denominator by 4:
$$\frac{\cancel{4}^{1}}{\cancel{324}^{81}} + 2 = \frac{1}{81} + 2$$
6. **Final answer:**
$$\frac{d^2y}{dx^2} \bigg|_{x=6} = 2 + \frac{1}{81} = \frac{162}{81} + \frac{1}{81} = \frac{163}{81}$$