Word Count : 645

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To apply the Implicit Function Theorem and solve the given problem, we need to express the situation in a way that satisfies the conditions of the theorem. ### Problem Recap: We are given the function \( F(x, y) = x^5 + y^5 - 16xy^3 - 1 \), and we are tasked with finding a function \( g(y) \) that satisfies \( F(g(y), y) = 0 \) for a neighborhood of \( y = 2 \), with the initial condition \( g(1) = 2 \). Additionally, we are asked to find \( g'(y) \). ### Step 1: Verify the conditions of the Implicit Function Theorem The Implicit Function Theorem states that if we have an equation of the form \( F(x, y) = 0 \), then, under suitable conditions, there exists a differentiable function \( g(y) \) that implicitly defines \( x \) in terms of \( y \), provided that: 1. \( F(x_0, y_0) = 0 \), 2. The partial derivative ____ ____ _____ __________ __________ _________.
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