# Multivariable chain rule

• October 8th 2009, 08:30 PM
theowne
Multivariable chain rule
Could I get some guidance on this subject?

Yfrog - vh5.jpg

Yfrog - dsc04451z

I thought I could do the first one by starting from the right. I figured I could replace du/ds with du/dx (dx/dt), replace dx/dt with the derivative given by the info, get e^2s and sin^2 + cos ^2 as a multiplier and end up with the left side. However, du/ds leaves cost as cost and only du/dt gives me a cos^2 so I don't get that clean 1. Am I missing something, made a great mistake, or just approaching this entirely wrongly?

• October 9th 2009, 06:43 AM
HallsofIvy
Quote:

Originally Posted by theowne
Could I get some guidance on this subject?

Yfrog - vh5.jpg

Yfrog - dsc04451z

I thought I could do the first one by starting from the right. I figured I could replace du/ds with du/dx (dx/dt), replace dx/dt with the derivative given by the info, get e^2s and sin^2 + cos ^2 as a multiplier and end up with the left side. However, du/ds leaves cost as cost and only du/dt gives me a cos^2 so I don't get that clean 1. Am I missing something, made a great mistake, or just approaching this entirely wrongly?

No, you can't "replace du/ds with du/dx (dx/dt)". $\frac{\partial u}{\partial s}= \frac{\partial f}{\partial x}\frac{\partial x}{\partial x}+ \frac{\partial f}{\partial y}\frac{\partial y}{\partial s}$