# Find the derivative of y= (1+√t)/(1-√t)?

`y = (1+sqrtt)/(1-sqrtt)`

`y = (1+sqrtt)/(1-sqrtt)*(1+sqrtt)/(1+sqrtt)`

`y = (1-sqrtt)^2/(1-(sqrtt)^2)`

`y = (1-sqrtt)^2/(1-t)`

(dy)/dt

`= ((1-t)(2(1+sqrtt)*(1/(2sqrtt)))-(1+sqrtt)^2(-1)}/(1-t)^2`

`= ((1-t)(1/sqrtt+1)+1+2sqrtt+t)/(1-t)^2`

`= (1/sqrtt+1-sqrtt-t+1+2sqrtt+t)/(1-t)^2`

`= (2+sqrtt+1/sqrtt)/(1-t)^2`

`= (sqrtsqrtt+1/sqrtsqrtt)^2/(1-t)^2`

`= (sqrtt+1)^2/(sqrtt(1-t)^2)`

Therefore `(dy)/dt = (sqrtt+1)^2/(sqrtt(1-t)^2)`

Approved by eNotes Editorial Team

Use the division fule for differentation:

If y = f(t)/g(t)

dy/dt = (f'(t)g(t)- f(t)g'(t))/(g(t))^2

Also, multiply the top and bottom by (1 + sqrt(t)) first

y = (1 + 2*sqrt(t) + t)/(1 - t)

so f(t) = (1 + sqrt(t))^2 = 1 + 2*sqrt(t) +t  = 1 +2*t^(1/2) + t

g(t) = 1 -t

Gives f'(t) = 2*1/2*t^(-1/2) + 1 = 1/sqrt(t) + 1

g'(t) = -1

So dy/dt = ((1/sqrt(t) + 1)*(1-t) - (1 + sqrt(t))^2*(-1))/(1-t)^2

= ((1/sqrt(t) + 1)*(1-t) + (1 + sqrt(t))^2)/(1-t)^2

= (1/sqrt(t) + 1)/(1-t) + ((1+sqrt(t))/(1-t))^2

Approved by eNotes Editorial Team
{"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "url": "https://www.enotes.com/homework-help/find-derivative-y-1-t-1-t-368707", "fetch_time": 1623534515000000000, "content_mime_type": "text/html", "warc_filename": "crawl-data/CC-MAIN-2021-25/segments/1623487586390.4/warc/CC-MAIN-20210612193058-20210612223058-00163.warc.gz", "warc_record_offset": 703611866, "warc_record_length": 18520, "token_count": 529, "char_count": 1057, "score": 4.125, "int_score": 4, "crawl": "CC-MAIN-2021-25", "snapshot_type": "latest", "language": "en", "language_score": 0.3760571777820587, "file_path": "/sailhome/yjruan/project/latent-lm/data/processed/finemath-4plus/finemath-4plus/train-00029-of-00064.parquet"}