Exercises

01

Partial derivatives of a composite

Let $f(x,y)=\sin (\frac{x}{1+y})$. Compute $\frac{\partial f}{\partial x}$ and $\frac{\partial f}{\partial y}$.

02

Evaluate partial with four variables

Let

$$f(x,y,z,w)=\frac{e^{xz+y}}{z^2+w}.$$

Calculate $f_z(0,0,1,1)$.

03

Higher order partials

Let $f(x,y,z)=x^3+y^2{e}^x$.

• (a) Calculate the second-order partial derivatives of $f$.
• (b) Calculate $f_{xyy}$.

04

Clairaut: choose the order wisely!

Let $f(x,y,z,w)=x^3{w}^2{z}^2+\sin (\frac{xy}{z^2})$. Compute $f_{zzwx}$ in your head. (Explain it on paper!)