Differentiating Complex Implicit Equations: A Comprehensive Guide

This is a good example of a challenging A-level-equivalent question that tests your proficiency in calculus. While it might seem daunting at first, with the right tools, it becomes quite manageable. In this article, we'll explore how to solve such problems using implicit differentiation and the chain rule. Let's dive in!

Problem Statement

Consider the equation: [xy^4 - x - y^2 x^4y^4.]

We aim to find the derivative (frac{dy}{dx}).

Solution

Let's start by breaking down the problem into manageable steps and using implicit differentiation to find (frac{dy}{dx}).

Method A: Expand and Differentiate

Expand the equation [xy^4 - x - y^2 x^4y^4] Differentiate each term implicitly with respect to (x): [frac{d}{dx}[xy^4] xcdotfrac{d}{dx}[y^4] y^4cdotfrac{d}{dx}[x],] [frac{d}{dx}[-x] -1,] [frac{d}{dx}[-y^2] -2ycdotfrac{dy}{dx},] [frac{d}{dx}[x^4y^4] x^4cdotfrac{d}{dx}[y^4] y^4cdotfrac{d}{dx}[x^4].] Rearrange to isolate (frac{dy}{dx}): [xy^4cdot(4ycdotfrac{dy}{dx}) y^4 - 2ycdotfrac{dy}{dx} x^4(y^4cdot4yfrac{dy}{dx} 4x^3) -1 0 - 1cdotfrac{dy}{dx} (x^4y^4cdot4frac{dy}{dx}).] Combine like terms and solve for (frac{dy}{dx}): [4xy^5cdotfrac{dy}{dx} y^4 - 2ycdotfrac{dy}{dx} 4x^4y^5frac{dy}{dx} 4x^7 -1 - frac{dy}{dx} 4x^4y^4frac{dy}{dx}.] After simplifying, we get the final answer:

Method B: Direct Differentiation

Alternatively, you can differentiate the entire equation directly without expanding:

Differentiate both sides with respect to (x): [frac{d}{dx}[xy^4 - x - y^2] frac{d}{dx}[x^4y^4].] Note that (frac{d}{dx}[xy^4]) requires the product rule and the chain rule: Let (u xy^4 Rightarrow frac{du}{dx} y^4 4xy^3frac{dy}{dx}.) (frac{d}{dx}[x^4y^4] 4x^3y^4 x^4cdot4y^3frac{dy}{dx}.) Rearrange and solve for (frac{dy}{dx}): [y^4 4xy^3frac{dy}{dx} - 1 - 2yfrac{dy}{dx} 4x^3y^4 4x^4y^3frac{dy}{dx}.] Combine like terms and isolate (frac{dy}{dx}): [4xy^3frac{dy}{dx} - 2yfrac{dy}{dx} - 4x^4y^3frac{dy}{dx} 4x^3y^4 - y^4 1.] After simplifying, we get the final answer:

Conclusion

Differentiation can be a powerful tool, and mastering techniques like implicit differentiation and the chain rule can significantly enhance your problem-solving skills in calculus. The key to success is plenty of practice with various types of problems, including both straightforward and scenario-based questions. So, keep practicing and stay curious!