Question

Find dy/dx in terms of x and y if x^(1/3)=2sqrt(y)

Answer 1

\[x \sqrt[3]{x}=2\sqrt{y}\]

Answer 2

you can just solve this for y and directly differentiate... no real need for implicit differentiation

Answer 3

\[\sqrt[3]{x ^{4}}=2\sqrt{y}\]

Answer 4

\[x ^{4/3}=2y ^{1/2}\]

Answer 5

\[y ^{1/2}=.5x ^{4/3}\]

Answer 6

\[y=(.5x ^{4/3})^{1/2}\]

Answer 7

\[y=.5^{1/2}x ^{2/3}\]

Answer 8

since the ?says ito of y and x so implicit is important
\[x^{1/3}=2\sqrt{y}\]
\[\frac{ 1 }{ 3x^{2/3} }=\frac{- y' }{ \sqrt{y} }\]
\[y'=-\frac{ \sqrt{y} }{ 3\sqrt[3]{x^2} }\]
in terms of x and y