Question

Solve the following:

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

Answer 1

`"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"`