ax...............

if y = ax prove that dy/dx = y2(log y)/x{1 – y(logx)(logy)}

explain in great detail

 

Asked by haroonrashidgkp | 13th Aug, 2018, 08:40: PM

Expert Answer:

begin mathsize 16px style straight y equals straight a to the power of straight x to the power of straight a to the power of straight x....... end exponent end exponent end exponent
straight y equals open parentheses straight a to the power of straight x close parentheses to the power of straight y
log space straight y equals straight x to the power of straight y log space straight a
Taking space log space on space both space sides
log left parenthesis log space straight y right parenthesis equals log space straight x to the power of straight y space plus space log left parenthesis log space straight a right parenthesis
log left parenthesis log space straight y right parenthesis equals ylog space straight x space plus space log left parenthesis log space straight a right parenthesis
fraction numerator 1 over denominator ylog space straight y end fraction dy over dx equals straight y cross times 1 over straight x plus log space straight x dy over dx
fraction numerator 1 over denominator straight y log space straight y end fraction dy over dx minus log space straight x dy over dx equals straight y over straight x
dy over dx open parentheses fraction numerator 1 over denominator ylog space straight y end fraction minus log space straight x close parentheses equals straight y over straight x
dy over dx open parentheses fraction numerator 1 minus ylog space straight y space log space straight x over denominator straight y log space straight y end fraction close parentheses equals straight y over straight x
dy over dx equals fraction numerator straight y squared log space straight y over denominator straight x left parenthesis 1 minus ylog space straight y space log space straight x right parenthesis end fraction end style

Answered by Sneha shidid | 16th Aug, 2018, 10:04: AM