Problems

01

Partial derivatives

• (a) Find $\frac{\partial }{\partial y}(x^2+y)(x+y^4)$.
• (b) Find $\frac{\partial }{\partial u}\ln (u^2+uv)$.
• (c) Compute $f_{xyxzy}$ for $f(x,y,z)=y\sin (xz)\sin (x+z)+(x+z^2)\tan (y)+x\tan \left(\frac{z+z^{-1}}{y-y^{-1}}\right)$.
  (Hint: you can use a different order on each term!)
• (d) The Laplace Operator $\Delta$ is defined by $\Delta f=f_{xx}+f_{yy}$. It is very important! Show that $u(x,y)=e^x\cos (y)$ and $v(x,y)=\ln (x^2+y^2)$ and $\theta (x,y)={\tan }^{-1}(\frac{y}{x})$ are all annihilated (sent to 0) by $\Delta$.