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Q. 23

Expert-verifiedFound in: Page 964

Book edition
1st

Author(s)
Peter Kohn, Laura Taalman

Pages
1155 pages

ISBN
9781429241861

Use Theorem 12.32 to find the indicated derivatives in Exercises 21–26. Express your answers as functions of a single variable. $\frac{dx}{dt}whenx=r\mathrm{cos}\theta ,r={t}^{2}-5,and\theta ={t}^{3}+1$

The value is $\frac{dx}{dt}=2t\xb7\mathrm{cos}({t}^{3}+1)-3{t}^{2}({t}^{2}-5)\mathrm{sin}({t}^{3}+1)$

Given:

$x=r\mathrm{cos}\theta ,\phantom{\rule{0ex}{0ex}}r={t}^{2}-5,and\phantom{\rule{0ex}{0ex}}\theta ={t}^{3}+1$

We have to find the indicated derivatives and express your answers as functions of a single variable.

$U\mathrm{sin}gr={t}^{2}-5and\theta ={t}^{3}+1inx=r\mathrm{cos}\theta weget\phantom{\rule{0ex}{0ex}}x=({t}^{2}-5)\xb7\mathrm{cos}({t}^{3}+1)\phantom{\rule{0ex}{0ex}}Diff.w.r.t.tweget\phantom{\rule{0ex}{0ex}}\frac{dx}{dt}=({t}^{2}-5)\frac{d}{dt}\mathrm{cos}({t}^{3}+1)+\mathrm{cos}({t}^{3}+1)\frac{d}{dt}({t}^{2}-5)\phantom{\rule{0ex}{0ex}}\frac{dx}{dt}=({t}^{2}-5)(-\mathrm{sin}({t}^{3}+1)\xb73{t}^{2})+\mathrm{cos}({t}^{3}+1)\xb72t\phantom{\rule{0ex}{0ex}}\frac{dx}{dt}=2t\xb7\mathrm{cos}({t}^{3}+1)-3{t}^{2}({t}^{2}-5)\mathrm{sin}({t}^{3}+1)$

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