Calculus Help
posted by Maho on .
use logarithmic diff. to find the derivative of the function. Show steps please! so I can see how it is done. Thank you so much!
y=(e^(x)cos^(2)(x))/(x^(2)+x+1)

y = (e^x) (cosx)^2 (x^2 + x + 1)
take ln of both sides
ln y = ln e^x + ln (cosx)^2 + ln(x^2 + x + 1)
= x + 2 ln(cosx) + ln(x^2 + x + 1)
now differentiate
y' / y = 1 + 2(sinx/cosx) + (2x+1)/(x^2 + x + 1)
= 1  2tanx + (2x+1)/(x^2 + x + 1)
y' = y(1  2tanx + (2x+1)/(x^2 + x + 1))
= [ (e^x) (cosx)^2 (x^2 + x + 1) ] * [ 1  2tanx + (2x+1)/(x^2 + x + 1) ]
sure hope they don't expect us to simplify this 
log y = log (e^x) + log cos^2(x)  log(x^2+x+1)
log y = x + 2log cos x  log(x^2+x+1)
1/y y' = 1  2tanx  (2x+1)/(x^2+x+1)
y' = (1 + 2tanx + (2x+1)/(x^2+x+1)) * (e^x cos^2x)/(x^2+x+1)
Now, you can massage that for a few more steps, to get something that pleases you