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Question
If x = t . log t, y = tt, then show that `dy/dx - y = 0`.
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Solution
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 = tt
Taking logarithm of both sides,
log y = log tt
log y = t . logt
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`
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