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

If x7⋅y9=(x + y)16, then show that dydx=yx - Mathematics and Statistics

Advertisement Remove all ads
Advertisement Remove all ads
Sum

If `"x"^7*"y"^9 = ("x + y")^16`, then show that `"dy"/"dx" = "y"/"x"`

Advertisement Remove all ads

Solution

`"x"^7*"y"^9 = ("x + y")^16`

Taking logarithm of both sides, we get

log `"x"^7*"y"^9` = log `("x + y")^16`

∴ log `"x"^7 + log "y"^9 = 16 log ("x + y")` 

∴ 7 log x + 9 log y = 16 log (x + y)

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

`7(1/"x") + 9(1/"y") "dy"/"dx" = 16(1/("x + y")) "d"/"dx" ("x + y")`

∴ `7/"x" + 9/"y" "dy"/"dx" = 16/("x + y") (1 + "dy"/"dx")`

∴ `7/"x" + 9/"y" "dy"/"dx" = 16/("x + y") + 16/("x + y") "dy"/"dx"`

∴ `9/"y" "dy"/"dx" - 16/("x + y") "dy"/"dx" = 16/("x + y") - 7/"x"`

∴ `(9/"y" - 16/("x + y")) "dy"/"dx" = 16/("x + y") - 7/"x"`

∴ `[("9x" + "9y" - 16"y")/("y"("x + y"))] "dy"/"dx" = (16"x" - 7"x" - 7"y")/("x"("x + y"))`

∴ `[("9x" - 7"y")/("y"("x + y"))] "dy"/"dx" = ("9x" - 7"y")/("x"("x + y"))`

∴ `"dy"/"dx" = ("9x" - 7"y")/("x"("x + y")) xx ("y"("x + y"))/("9x" - 7"y")`

∴ `"dy"/"dx" = "y"/"x"`

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

APPEARS IN

Balbharati Mathematics and Statistics 1 (Commerce) 12th Standard HSC Maharashtra State Board
Chapter 3 Differentiation
Miscellaneous Exercise 3 | Q 4.13 | Page 100
Advertisement Remove all ads
Share
Notifications

View all notifications


      Forgot password?
View in app×