If x^m * y^n = (x + y)^(m + n). Then how will you prove that dy/dx = y/x??
1 Answer
This is very similar to this problem:
https://socratic.org/questions/what-is-the-derivative-of-x-y-mn-x-m-y-n
Here, we have:
(x+y)^(m+n) = x^my^n
Taking Natural logarithms of both sides:
ln {(x+y)^(m+n)} = ln{ x^my^n}
And using the rules of logs we can manipulate as follows:
(m+n)ln (x+y) = ln x^m + lny^n
:. (m+n)ln (x+y) = mln x + nlny
Differentiate Implicitly:
(m+n) * 1/(x+y) * (1+y') = m/x + n/y * y'
:. (m+n) (1+y')/(x+y) = (my + nxy')/(xy)
Cross multiply by
(m+n) (1+y')xy = (my + nxy')(x+y)
Multiply out:
mxy+mxyy' + nxy+nxyy' = mxy + nx^2y' + my^2 + nxyy'
:. mxyy' + nxy = nx^2y' + my^2
Finally, Collect term,s and tidy up:
mxyy' -nx^2y' = my^2 - nxy
:. (my -nx)xy' = (my - nx)y
:. xy' = y
:. y' = y/x QED
This assumes that where applicable the denominators are non-zero to permit the division to occur.