If xmyn= (x+y)m+n, prove that dy/dx = y/x

Mention each and every step

Asked by haroonrashidgkp | 19th Jun, 2018, 11:02: PM

Expert Answer:

xmyn= (x+y)m+n
logxm + logy
begin mathsize 16px style straight m over straight x plus straight n over straight y fraction numerator begin display style dy end style over denominator begin display style dx end style end fraction equals fraction numerator straight m plus straight n over denominator straight x plus straight y end fraction open parentheses 1 plus dy over dx close parentheses
fraction numerator begin display style straight m end style over denominator begin display style straight x end style end fraction plus fraction numerator begin display style straight n end style over denominator begin display style straight y end style end fraction fraction numerator begin display style dy end style over denominator begin display style dx end style end fraction equals fraction numerator begin display style straight m plus straight n end style over denominator begin display style straight x plus straight y end style end fraction plus fraction numerator begin display style straight m plus straight n end style over denominator begin display style straight x plus straight y end style end fraction fraction numerator begin display style dy end style over denominator begin display style dx end style end fraction
fraction numerator begin display style straight n end style over denominator begin display style straight y end style end fraction fraction numerator begin display style dy end style over denominator begin display style dx end style end fraction minus fraction numerator begin display style straight m plus straight n end style over denominator begin display style straight x plus straight y end style end fraction fraction numerator begin display style dy end style over denominator begin display style dx end style end fraction equals fraction numerator begin display style straight m plus straight n end style over denominator begin display style straight x plus straight y end style end fraction minus fraction numerator begin display style straight m end style over denominator begin display style straight x end style end fraction
open parentheses fraction numerator begin display style straight n end style over denominator begin display style straight y end style end fraction minus fraction numerator begin display style straight m plus straight n end style over denominator begin display style straight x plus straight y end style end fraction close parentheses fraction numerator begin display style dy end style over denominator begin display style dx end style end fraction equals fraction numerator mx plus nx minus mx minus my over denominator straight x left parenthesis straight x plus straight y right parenthesis end fraction
open parentheses fraction numerator nx plus ny minus my minus ny over denominator left parenthesis straight x plus straight y right parenthesis straight y end fraction close parentheses fraction numerator begin display style dy end style over denominator begin display style dx end style end fraction equals fraction numerator mx plus nx minus mx minus my over denominator straight x left parenthesis straight x plus straight y right parenthesis end fraction
fraction numerator nx minus my over denominator straight y end fraction fraction numerator begin display style dy end style over denominator begin display style dx end style end fraction equals fraction numerator begin display style nx minus my end style over denominator begin display style straight x end style end fraction
fraction numerator begin display style dy end style over denominator begin display style dx end style end fraction equals straight y over straight x end stylen = (m+n)log(x+y)
mlogx + nlogy = (m+n)log(x+y)

Answered by Sneha shidid | 20th Jun, 2018, 10:10: AM