If y^3=9x^2, determine `dx/dt` when x=3 and `dy/dt` =3

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justaguide | College Teacher | (Level 2) Distinguished Educator

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The derivative `dx/dt` has to be determined given that `y^3=9x^2` when `x=3` and `dy/dt=3` .

`y^3 = 9x^2`

Use implicit differentiation to differentiate both the sides with respect to t.

`3*y^2*(dy/dt) = 18*x*(dx/dt)`

=> `dx/dt = (y^2/(6x))*(dy/dt)`

When x = 3 and `dy/dt = 3` , `dx/dt` has to be determined.

First determine y, `y^3 = 9*9 = 81`

y = `3*root(3) 3`

`dx/dt = ((9*3^(2/3))/18)*3`

= `(3/2)*3^(2/3)`

The required value of `dx/dt = (3/2)*3^(2/3)`

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