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प्रश्न
If x = t . log t, y = tt, then show that `"dy"/"dx" - "y" = 0`
उत्तर
x = t . log t
Differentiating both sides w.r.t. t
`dx/dt=td/dt(logt)+logtd/dt(t)`
= `txx1/t+logt(1)`
`dx/dt=1+logt` ...(i)
y = tt
Taking logarithm of both sides
logy = logtt
logy = t.logt
Differentiating both sides w.r.t. t
`1/yxxdy/dt=td/dt(logt)+logtd/dt(t)`
= `txx1/t+logt(1)`
= 1 + logt
`dy/dt=y(1+logt)` ...(ii)
`dy/dx=(dy/dt)/(dx/dt)`
= `(y(1+logt))/((1+logt))` ...[From (i) & (ii)]
∴ `dy/dx=y`
∴ `dy/dx-y=0`
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