Maharashtra State BoardHSC Commerce 12th Board Exam
Advertisement Remove all ads

Find dydx if, xy=ex - y - Mathematics and Statistics

Advertisement Remove all ads
Advertisement Remove all ads
Sum

Find `"dy"/"dx"` if, `"x"^"y" = "e"^("x - y")`

Advertisement Remove all ads

Solution

`"x"^"y" = "e"^("x - y")`

Taking logarithm of both sides, we get

y log x = (x - y) log e = x - y

∴ y log x + y = x

∴ y(1 + log x) = x

∴ y = `"x"/(1 + log "x")`

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

`"dy"/"dx" = "d"/"dx" ["x"/(1 + log "x")]`

∴ `"dy"/"dx" = ((1 + log "x") "d"/"dx" ("x") - "x" "d"/"dx" (1 + log "x"))/(1 + log "x")^2`

`= ((1 + log "x") xx 1 - "x" xx (1/"x"))/(1 + log "x")^2`

`= (1 + log "x" - 1)/(1 + log "x")^2`

∴ `"dy"/"dx" = (log "x")/(1 + log "x")^2`

Concept: Derivatives of Implicit Functions
  Is there an error in this question or solution?
Advertisement Remove all ads

APPEARS IN

Advertisement Remove all ads
Share
Notifications

View all notifications


      Forgot password?
View in app×