please explain the steps of using implicit differentiation to obtain: `(da)/(dt)` of `a^4-t^4=5a^2t`

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The derivative `(da)/(dt)` has to be determined given that `a^4 - t^4 = 5*a^2*t`

Take the derivative of both the sides with respect to t

`(d(a^4 - t^4))/(dt) = (d(5*a^2*t))/(dt)`

=> `4*a^3*((da)/dt) - 4t^3 = 5*(2a*((da)/dt)*t + a^2)`

=> `4*a^3*((da)/dt) - 4t^3 = 10a*((da)/dt)*t + 5a^2`

=> `((da)/(dt))(4a^3 - 10at) = 5a^2 + 4t^3`

=> `(da)/(dt) = (5a^2 + 4t^3)/(4a^3 - 10at)`

**The required derivative **`(da)/dt = (5a^2 + 4t^3)/(4a^3 - 10at)`

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