Let G(x, r, t) = 2 $x ^ { 2 }$ t + $\frac { 1 } { 3 }$ $t ^ { 2 }$ - $r ^ { 3 }$ $\sqrt { x t }$ . Find $\frac { \partial G } { \partial x }$ .
A) 4xt - $\frac { 1 } { 3 }$ $t ^ { 2 }$ - $\frac { 3 \mathrm { r } ^ { 2 } \mathrm { t } } { 2 \sqrt { x t } }$
B) 4x - $\frac { \mathrm { r } ^ { 3 } } { 2 \sqrt { \mathrm { xt } } }$
C) $4 x t - \frac { r ^ { 3 } t } { 2 \sqrt { x t } }$
D) 4xt + $\frac { 1 } { 3 }$ $t ^ { 2 }$ - $r ^ { 3 }$ $\frac { 1 } { 2 \sqrt { x t } }$
E) none of these

Question

Quizplus · Accepted Answer

To find the partial derivative of $G(x, r, t)$ with respect to $x$, we differentiate each term containing $x$ while treating other variables as constants. The differentiation of $2x^2t$ with respect to $x$ gives $4xt$, and the differentiation of $-r^3\sqrt{xt}$ with respect to $x$ gives $-\frac{r^3t}{2\sqrt{xt}}$, leading to the correct answer: $4xt - \frac{r^3t}{2\sqrt{xt}}$.

Let G(x, R, T) = 2 x2x ^ { 2 }x2

Let G(x, R, T) = 2 $x ^ { 2 }$