If y^x = e^(y-x) then prove dy/dx = (1+logy)^2/logy

Asked by ranaakshat345 | 17th Jun, 2019, 01:28: PM

Expert Answer:

begin mathsize 16px style straight y to the power of straight x space equals space straight e to the power of straight y minus straight x end exponent
Taking space log space on space both space the space sides comma space we space get
log straight y to the power of straight x space equals space loge to the power of straight y minus straight x end exponent
rightwards double arrow xlogy space equals space open parentheses straight y space minus space straight x close parentheses loge
rightwards double arrow xlogy space equals space straight y space minus space straight x space space space space space..... space left parenthesis 1 right parenthesis
Differenciating space straight w. straight r. straight t space straight x space comma space we space get
straight x space straight d over dx open parentheses logy close parentheses space plus space logy space straight d over dx open parentheses straight x close parentheses space equals space straight d over dx open parentheses straight y close parentheses space minus space straight d over dx open parentheses straight x close parentheses
rightwards double arrow space straight x space cross times space 1 over straight y cross times space dy over dx space plus space logy space equals space dy over dx space minus space 1 space
rightwards double arrow space dy over dx space equals space fraction numerator 1 space plus space logy over denominator 1 space minus space begin display style straight x over straight y end style end fraction
rightwards double arrow space dy over dx space equals space fraction numerator straight y open parentheses 1 space plus space logy close parentheses over denominator straight y space minus space straight x end fraction
space space space space space space space space space space space space equals fraction numerator straight x open parentheses 1 space plus space logy close parentheses squared over denominator xlogy end fraction
space space space space space space space space space space space space equals open parentheses 1 space plus space logy close parentheses squared over logy space
Hence space proved.
space
space space end style

Answered by Yasmeen Khan | 17th Jun, 2019, 02:53: PM

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