• :00Days
• :00Hours
• :00Mins
• 00Seconds
A new era for learning is coming soon

Suggested languages for you:

Americas

Europe

TF. 4

Expert-verified
Found in: Page 946

### Calculus

Book edition 1st
Author(s) Peter Kohn, Laura Taalman
Pages 1155 pages
ISBN 9781429241861

# $\mathrm{A}\mathrm{chain}\mathrm{rule}\mathrm{for}\mathrm{functions}\mathrm{of}\mathrm{two}\mathrm{variables}:\mathrm{Let}f\left(x,y\right)={x}^{2}{y}^{3},x=s\mathrm{cos}t,\mathrm{and}y=s\mathrm{sin}t.\phantom{\rule{0ex}{0ex}}\mathrm{Substitute}\mathrm{the}\mathrm{expressions}\mathrm{for}x\mathrm{and}y\mathrm{into}f\left(x,y\right).$

$f\left(x,y\right)={s}^{5}{\mathrm{cos}}^{2}t{\mathrm{sin}}^{3}t$

See the step by step solution

## Step 1. Given information

$f\left(x,y\right)={x}^{2}{y}^{3},x=s\mathrm{cos}t,\mathrm{and}y=s\mathrm{sin}t.$

## Step 2. Substitute the expressions for x and y into f(x, y)

$f\left(x,y\right)={x}^{2}{y}^{3},x=s\mathrm{cos}t,\mathrm{and}y=s\mathrm{sin}t\phantom{\rule{0ex}{0ex}}⇒f\left(x,y\right)={\left(s\mathrm{cos}t\right)}^{2}×{\left(s\mathrm{sin}t\right)}^{3}\phantom{\rule{0ex}{0ex}}⇒f\left(x,y\right)=\left({s}^{2}{\mathrm{cos}}^{2}t\right)×\left({s}^{3}{\mathrm{sin}}^{3}t\right)\phantom{\rule{0ex}{0ex}}⇒f\left(x,y\right)={s}^{5}{\mathrm{cos}}^{2}t{\mathrm{sin}}^{3}t$