How Do You Tackle Partial Differentiation with Trigonometric Functions?

[phrygian]

Homework Statement

If x = yz and y = 2sin(y+z), find dx/dy and d^2x/dy^2

Homework Equations

The Attempt at a Solution

I am beyond confused at how to even start this one, the problem is not like any of the examples in my book.

I know x = ydz + zdy but don't know how to deal with z and dz?

Thanks for the help

 

[phsopher]

Find dz in terms of dy from the second equation and plug it into the first: dx=ydz+zdy.

 

[phrygian]

Is it correct to use inverse trig function for that?

 

[HallsofIvy]

I can see no reason to. The second equation is y= 2sin(y+z) so dy= 2cos(y+z)(dy+ dz). From dx= zdy+ ydz, (NOT the "x= zdy+ ydz" you give. I assume that was a typo.), we get ydz= dx- zdy so dz= (1/y)dx- (z/y)dy. Putting that into the second equation, dy= 2cos(y+z)(dy+ (1/y)dx- (z/y)dy)= 2cos(y+z)(1- (z/y)dy+ (2/y)cos(y+z)dx. Solve that for dx and divide both sides by dy.

 

[Mindscrape]

You can do anything you want that doesn't violate the laws of math!