$\text { Find } \frac { \partial w } { \partial t } \text { using the appropriate Chain Rule for } w = y ^ { 3 } - 8 x ^ { 2 } y \text { where } x = e ^ { 3 } \text { and }$ $y = e ^ { t }$, and evaluate the partial derivative at $s = - 3$ and $t = 2$. Round your answer to two decimal places.
A) $1,209.99$
B) $806.71$
C) $1,210.14$
D) $806.69$
E) $1,209.20$

Question

Quizplus · Accepted Answer

To find $\frac{\partial w}{\partial t}$ using the chain rule, we first express $w$ in terms of $t$ using $x = e^3$ (which is a constant) and $y = e^t$. Thus, $w = (e^t)^3 - 8(e^3)^2(e^t) = e^{3t} - 8e^6e^t$. Taking the derivative of $w$ with respect to $t$ gives $\frac{\partial w}{\partial t} = 3e^{3t} - 8e^6e^t$. Evaluating this at $t = 2$ gives $3e^6 - 8e^8$. Substituting $e^2 \approx 7.39$ (since $e \approx 2.718$), we get $3(7.39)^3 - 8(7.39)^4$, which when calculated gives approximately $1210.14$.

y=ety = e ^ { t }y=et , and Evaluate the Partial Derivative At

$y = e ^ { t }$ , and Evaluate the Partial Derivative At