Question

If x^a .y^b = `(x + y)^((a + b))`, then show that `("d"y)/("d"x) = y/x`

Answer 1

x^a .y^b = `(x + y)^((a + b))`

Taking logarithm of both sides, we get

log(x^a.y^b) = log(x + y)^a+b 

∴ log x^a + log y^b = (a + b) log (x + y)

∴ a log x + b log y = (a + b) log (x + y)

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

`"a"*1/x + "b"*1/y*("d"y)/("d"x) = ("a" + "b")*1/(x + y)*"d"/("d"x)(x + y)`

∴ `"a"/x + "b"/y*("d"y)/("d"x) = ("a" + "b")/(x + y)(1 + ("d"y)/("d"x))`

∴ `"a"/x + "b"/y*("d"y)/("d"x) = ("a" + "b")/(x + y) + ("a" + "b")/(x + y) . ("d"y)/("d"x)`

∴ `"b"/y*("d"y)/("d"x) - ("a" + "b")/(x + y)*("d"y)/("d"x) = ("a" + "b")/(x + y) - "a"/x` 

∴ `("b"/y - ("a" + "b")/(x + y)) ("d"y)/("d"x) = ("a" + "b")/(x + y) - "a"/x`

∴ `[("b"x + "b"y - "a"y - "b"y)/(y(x + y))]("d"y)/("d"x) = ("a"x + "b"x - "a"x - "a"y)/(x(x + y))`

∴ `[("b"x - "a"y)/(y(x + y))]("d"y)/("d"x) = ("b"x - "a"y)/(x(x + y))`

∴ `("d"y)/("d"x) = ("b"x - "a"y)/(x(x + y)) xx (y(x + y))/("b"x - "a"y)`

∴ `("d"y)/("d"x) = y/x`