Chapter 14 - Partial Derivatives - 14.3 Partial Derivatives - 14.3 Exercises - Page 964: 16

$$f_x = -t^2e^{-x}.$$ $$f_t = 2te^{-x}.$$

To find $f_x$, we treat $t$ as a constant and differentiate with respect to $x$. Thus $$f_x = t^2 \cdot -e^{-x} = -t^2e^{-x}.$$ Similarly, to find $f_t$, we treat $x$ as constant and differentiate with respect to $t$. Thus $$f_t = 2te^{-x}.$$