Aerospace Engineering (Structural Integrity under Thermal Expansion)

A structural aerospace engineer is inspecting a circular optical sensor viewing port integrated into the titanium forward bulkhead of a crew exploration vehicle. Due to aerodynamic friction during atmospheric re-entry, uneven thermal expansion deforms the window mounting bezel along an implicit circle:

where $x$ and $y$ are coordinate displacements in centimeters. A stress concentration sensor focuses on the perimeter point $(x, y) = (3, 4)$.

1. Compute the first derivative $\frac{dy}{dx}$ using implicit differentiation.
2. Differentiate $\frac{dy}{dx}$ a second time with respect to $x$ using the Quotient Rule to establish an expression for the second derivative $\frac{d^2y}{dx^2}$.
3. Back-substitute $\frac{dy}{dx}$ and simplify the numerator using the original circle constraint equation ($x^2 + y^2 = 25$).
4. Evaluate $\frac{d^2y}{dx^2}$ at the physical inspection point $(3, 4)$, and determine whether the window bezel is concave up or concave down.