If Z=x^3cos(y/x), find Zx and Zy. Hence Show that XZx + YZy=3Z

Question

Also falls under Functions of two variables if i am not mistaken.

Answer 1

Hey Teddy.

 

Zx and Zy are partial derivatives. Zx means you take the derivative of Z wherein x is your only variable, leaving y to be treated as a constant. Zy means x is your constant and y is now your variable. As you can see from your equation, Zx is going to be substantially more complicated.

 

Let's start with Zy. You'll leave the x³ term out front alone since it's a constant, and then employ the chain rule with sin.

 

Z = x³*cos(y*1/x)

 

Zy = x³*(-sin(y*1/x)*1/x)

= -x²*sin(y/x)

 

Now Zx. You'll have to employ both the product rule and the chain rule for this one.

 

Zx = 3*x²*cos(y/x) + x³*(-sin(y/x)*y*(-1/x²))

 

= 3*x²*cos(y/x) + y*x*sin(y/x)

 

= 3x²*cos(y/x) + xy*sin(y/x)

 

Your question then says so show that x*Zx + y*Zy = 3Z

 

x*Zx = 3x³*cos(y/x) + x²*y*sin(y/x)

 

y*Zy = -x²*y*sin(y/x)

 

x*Zx + y*Zy = 3x³*cos(y/x) + x²*y*sin(y/x) - x²*y*sin(y/x)

 

The last two terms cancel, and you're left with just

 

3x³*cos(y/x), which is 3Z.

 

Hope this helps.