## Thomas' Calculus 13th Edition

The Derivative is: $r'=\frac{\theta-1}{\theta^{\frac{3}{2}}}$
$r=2(\frac{1}{\sqrt{\theta}}+\sqrt{\theta})$ $r=2(\frac{1\space+\space\theta}{\sqrt{\theta}})$ Using Quotient Rule to find the Derivative: $y'=\frac{f'(x)\cdot g(x)-f(x)\cdot g'(x)}{g^2(x)}$ $r'=2(\frac{(0+(1)\theta^{1-1})(\sqrt{\theta})-(1+\theta)((\frac{1}{2})\theta^{\frac{1}{2}-1})}{(\sqrt{\theta})^2})$ $r'=2(\frac{\sqrt{\theta}-(1+\theta)(\frac{1}{2\sqrt{\theta}})}{\theta})$ $r'=2(\frac{\sqrt{\theta}-\frac{1-\theta}{2\sqrt{\theta}})}{\theta}$ $r'=2(\frac{\frac{2\theta-1-\theta}{2\sqrt{\theta}}}{\theta})$ $r'=2(\frac{\theta-1}{2\theta\sqrt{\theta}})$ $r'=\frac{\theta-1}{\theta\sqrt{\theta}}$ $r'=\frac{\theta-1}{\theta^{\frac{3}{2}}}$
{"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "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.gradesaver.com/textbooks/math/calculus/thomas-calculus-13th-edition/chapter-3-derivatives-section-3-3-differentiation-rules-exercises-3-3-page-125/26", "fetch_time": 1596541218000000000, "content_mime_type": "text/html", "warc_filename": "crawl-data/CC-MAIN-2020-34/segments/1596439735867.93/warc/CC-MAIN-20200804102630-20200804132630-00179.warc.gz", "warc_record_offset": 689841056, "warc_record_length": 13535, "token_count": 320, "char_count": 740, "score": 3.890625, "int_score": 4, "crawl": "CC-MAIN-2020-34", "snapshot_type": "latest", "language": "en", "language_score": 0.28872042894363403, "file_path": "/sailhome/yjruan/project/latent-lm/data/processed/finemath-4plus/finemath-4plus/train-00005-of-00064.parquet"}