If y = tan-1 a/x + log (x-a/x+a)1/2, prove that dy/dx = 2a3/(x4 – a4)

Mention each and every step

Asked by haroonrashidgkp | 19th Jun, 2018, 11:00: PM

Expert Answer:

y = tan-1 a/x + log (x-a/x+a)1/2 
y = u + v
begin mathsize 16px style straight u equals tan to the power of negative 1 end exponent open parentheses straight a over straight x close parentheses
du over dx equals fraction numerator 1 over denominator 1 plus open parentheses begin display style straight a over straight x end style close parentheses squared end fraction cross times fraction numerator negative 1 over denominator straight x squared end fraction
du over dx equals fraction numerator negative 1 over denominator straight x squared plus straight a squared end fraction
straight v equals log open parentheses fraction numerator straight x minus straight a over denominator straight x plus straight a end fraction close parentheses to the power of 1 half end exponent
straight v equals 1 half log open parentheses fraction numerator straight x minus straight a over denominator straight x plus straight a end fraction close parentheses
straight v equals 1 half open square brackets log open parentheses straight x minus straight a close parentheses minus log open parentheses straight x plus straight a close parentheses close square brackets
dv over dx equals 1 half open parentheses fraction numerator 1 over denominator straight x minus straight a end fraction minus fraction numerator 1 over denominator straight x plus straight a end fraction close parentheses
dy over dx equals fraction numerator negative 1 over denominator straight x squared plus straight a squared end fraction plus 1 half open parentheses fraction numerator 1 over denominator straight x minus straight a end fraction minus fraction numerator 1 over denominator straight x plus straight a end fraction close parentheses end style

Answered by Sneha shidid | 20th Jun, 2018, 10:16: AM