ANSWERS: 2

dy/dx = (3x²)e^(x³) If you don't understand how I got that (the chain rule), then you might as well not read further. Now find the inverse function: x³ = ln(y); x = (ln(y))^1/3 Note that the domain of this function is restricted to values y>0. This is a composite function: x=f(g(y)) where f=g^1/3 and g=ln(y), so again use the chain rule: dx/dy = (1/3)((ln(y))2/3)(1/y) NOW substitute y = e^(x³)  the original equation in your posted question. so dx/dy = (1/3)(x³^2/3)(e^(x³)) = (1/3x²)(e^(x³)) Compare this to the first line of my answer: dy/dx = (3x²)e^(x³) They are reciprocals  their product is 1. qed :)

I think dy/dx is always the inverse of dx/dy, by definition? :) Just do the differentiation and you'll find it, though. See xprofessor above
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