Problem

2. (a) $f(x) = [\ln(1+x^2)]^x$. $\displaystyle\frac{d}{dx}f(x) = \underline{\quad(4)\quad}$.

(b) $x^3-y^2+y^3=x$. At $(x,y)=(0,1)$, $\displaystyle\frac{d^2y}{dx^2} = \underline{\quad(5)\quad}$.

(c) $f(x,y,z)=\displaystyle\int_z^{xy} e^{\sqrt{t}}\,dt$. $\nabla f = \underline{\quad(6)\quad}$.

(d) $f(x,y)=\displaystyle\frac{\sin(x^2y)}{x^2+y^2}$ for $(x,y)\ne(0,0)$ and $f(0,0)=0$. The directional derivative of $f$ along $\mathbf{u}=(\cos\theta,\sin\theta)$ at $(0,0)$ is $\underline{\quad(7)\quad}$.