\(\dfrac{\mathrm d}{\mathrm dt}\left[2x^2+y^3\right]=\dfrac{\mathrm d}{\mathrm dt}[10]\implies 4x\dfrac{\mathrm dx}{\mathrm dt}+3y^2\dfrac{\mathrm dy}{\mathrm dt}=0\)
When \(x=1\), the original equation tells you that you have
\(2\times1^2+y^3=10\implies y^3=8\implies y=2\)
You also know that when \(x=1\), you have \(\dfrac{\mathrm dy}{\mathrm dt}=3\). Substituting everything you know into the differentiated equation, you get
\(4\times1\times\dfrac{\mathrm dx}{\mathrm dt}+3\times2^2\times3=0\implies\dfrac{\mathrm dx}{\mathrm dt}=-9\)
So \(x\) is changing at a rate of \(-9\text{ units/min}\).
