# Solve the following: If x5⋅y7=(x + y)12 then show that, dydx=yx - Mathematics and Statistics

Sum

Solve the following:

If "x"^5 * "y"^7 = ("x + y")^12 then show that, "dy"/"dx" = "y"/"x"

#### Solution

"x"^5 * "y"^7 = ("x + y")^12

Taking logarithm of both sides, we get

log ("x"^5 * "y"^7) = log ("x + y")^12

∴ log x5 + log y7 = 12 log (x + y)

∴ 5 log x + 7 log y = 12 log (x + y)

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

5. 1/"x" + 7. 1/"y" * "dy"/"dx" = 12 * 1/("x + y") * "d"/"dx" ("x + y")

∴ 5/"x" + 7/"y" * "dy"/"dx" = 12/("x + y") [1 + "dy"/"dx"]

∴ 5/"x" + 7/"y" * "dy"/"dx" = 12/("x + y") + 12/("x + y") * "dy"/"dx"

∴ [7/"y" - 12/("x + y")] "dy"/"dx" = 12/("x + y") - 5/"x"

∴ [(7"x" + 7"y" - 12"y")/("y" ("x + y"))] "dy"/"dx" = (12"x" - 5"x" - 5"y")/("x"("x + y"))

∴ [("7x" - 5"y")/("y"("x + y"))] "dy"/"dx" = [("7x" - 5"y")/("x"("x + y"))]

∴ "dy"/"dx" = [("7x" - 5"y")/("x"("x + y"))] xx ("y"("x + y"))/("7x" - 5"y")

∴ "dy"/"dx" = "y"/"x"

Is there an error in this question or solution?
Chapter 3: Differentiation - Exercise 3.4 [Page 95]

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