If x = t.logt, y = tt, then show that dydx = tt - Mathematics and Statistics

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Sum

If x = t.logt, y = tt, then show that `("d"y)/("d"x)` = tt 

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Solution

x = t.logt    ......(i)

y = tt     ......(ii)

Taking logarithm of both sides, we get

log y = log tt

∴ log y = t.logt

∴ log y = x      ......[From (i)]

Differentiating both sides w.r.t. x, we get

`1/y*("d"y)/("d"x)` = 1

∴ `("d"y)/("d"x)` = y

∴ `("d"y)/("d"x)` = tt     ......[From (ii)]

Concept: The Concept of Derivative - Derivatives of Logarithmic Functions
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Chapter 1.3: Differentiation - Q.4
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