Question

If x = t . log t, y = t^t, then show that `dy/dx - y = 0`.

Answer 1

x = t . log t

Differentiating both sides w.r.t. t

`dx/dt = t d/dt (log t) + log t d/dt (t)`

= `t xx 1/t + log t (1)`

`dx/dt = 1 + log t`  ...(i)

y = t^t

Taking logarithm of both sides, 

log y = log t^t

log y = t . log^t

Differentiating both sides w.r.t. t

`1/y xx dy/dt = t d/dt (log t) + log t d/dt (t)`

= `t xx 1/t + log t (1)`

= 1 + log t

`dy/dt = y(1 + log t)`   ...(ii)

`dy/dx = (dy/dt)/(dx/dt)`

= `(y(1 + log t))/((1 + log t))`   ...[From (i) and (ii)]

∴ `dy/dx = y`

∴ `dy/dx - y = 0`