Answer
(a) \[(y^3ze^{xyz})\hat{i}+(e^{xyz}(2y+xy^2z))\hat{j}+(xy^2ze^{xyz})\hat{k}\]
(b) \[-\hat{i}+2\hat{j}\]
(c) \[\frac{5}{13}\]
Work Step by Step
It is given that \[f(x,y,z)=y^2e^{xyz}\]
Differentiate $f$ with respect to $x$ treating $y$ and $z$ constant.
\[f_x(x,y,z)=y^2e^{xyz}(yz)=y^3ze^{xyz}\]
Differentiate $f$ with respect to $y$ treating $x$ and $z$ constant.
\[f_y(x,y,z)=(2y)e^{xyz}+y^2e^{xyz}(xz)=e^{xyz}(2y+xy^2z) \]
Differentiate $f$ with respect to $z$ treating $x$ and $y$ constant.
\[f_z(x,y,z)=y^2e^{xyz}(xz)=xy^2ze^{xyz}\]
(a) \[\nabla f(x,y,z)=f_x(x,y,z)\hat{i}+f_y(x,y,z)\hat{j}+f_z(x,y,z)\hat{k}\]
\[\nabla f(x,y,z)=(y^3ze^{xyz})\hat{i}+(e^{xyz}(2y+xy^2z))\hat{j}+(xy^2ze^{xyz})\hat{k}\]
(b) \[\nabla f(0,1,-1)=f_x(0,1,-1)\hat{i}+f_y(0,1,-1)\hat{j}+f_z(0,1,-1)\hat{k}\]
\[\nabla f(0,1,-1)=-\hat{i}+2\hat{j}\]
(c) \[\mathbf{u}=\left(\frac{3}{13}\right)\hat{i}+\frac{4}{13}\hat{j}+\frac{12}{13}\hat{k}\]
Required directional derivative is given by:
\[D_{\mathbf{u}}f(0,1,-1)=\nabla f(0,1,-1)\cdot \mathbf{u}\]
\[D_{\mathbf{u}}f(0,1,-1)=(-\hat{i}+2\hat{j})\cdot \left(\frac{3}{13}\hat{i}+\frac{4}{13}\hat{j}+\frac{12}{13}\hat{k}\right)\]
\[D_{\mathbf{u}}f(0,1,-1)=\frac{5}{13}\]
