Final answer:
To rewrite the relation using the definition of inverse cosine and then find y/x in terms of x using implicit differentiation, one must first express the cosine of y in terms of the given expression and then differentiate both sides with respect to x. This involves isolating dy/dx and simplifying, understanding that y is a function implicitly defined by the relation given.
Step-by-step explanation:
To use the definition of inverse cosine to rewrite the given relation and then use implicit differentiation to find y/x in terms of x, we begin by understanding that the inverse cosine function, ¹⁺cos(x), is defined such that if y = ¹⁺cos(x), then x = cos(y). The given relation is ¹⁺cos¹(e²⁴ⁿ tan(x) - sin(x³)). If we let y equal this expression, then we have cos(y) = e²⁴ⁿ tan(x) - sin(x³). Using implicit differentiation, we differentiate both sides of the equation with respect to x. This yields -sin(y) * (dy/dx) = 2e²⁴ⁿ tan(x) + e²⁴ⁿ sec²(x) - 3x² cos(x³). We can then isolate dy/dx by dividing both sides of the equation by -sin(y), resulting in dy/dx = -(2e²⁴ⁿ tan(x) + e²⁴ⁿ sec²(x) - 3x² cos(x³))/sin(y). To find y/x, we simply divide dy by dx, which gives us the expression for dy/dx we just found. However, the precise form of y/x in terms of x depends on the context and what additional information we have about the function y and its relationship to x.