Question

find all of the first derivatives of f(x,y,z) =xsin(y-z)

Answer 1

\[ \frac{\partial}{\partial x}f(x,y,z)=\sin(y-z)\\
\frac{\partial}{\partial y}f(x,y,z)=\cos(y-z)\\
\frac{\partial}{\partial z}f(x,y,z)=\cos(y-z)(-1)=-\cos(y-z)\]
The last one requires the negative one since you find the derivative of the inner function (y-z), and with respect to z, you find the derivative on -z. (The same applies to the one above, but the derivative of )y-z) with respect to y is just 1)

Remember when you find the derivative with respect to say \(x\), then y and z re considered constants. So, in \(x\sin(y-z)\), just consider \(\sin(y-z)\) like a constant number, so you just worry about integrating the x-part, using the power rule. The same logic applies for the other derivatives

Answer 2

the last two have typos

Answer 3

hm yes I forgot the x's... oye..

Answer 4

\[\frac{\partial}{\partial x}f(x,y,z)=\sin(y-z)\\
\frac{\partial}{\partial y}f(x,y,z)=x\cos(y-z)\\
\frac{\partial}{\partial z}f(x,y,z)=x\cos(y-z)(-1)=-x\cos(y-z)\]