Find the indicated derivative.
-$\frac { \mathrm { dt } } { \mathrm { dx } }$ if $\mathrm { t } = \frac { \mathrm { x } } { 8 \mathrm { x } - 2 }$
A) $- \frac { 2 } { ( 8 x - 2 ) ^ { 2 } }$
B) $\frac { 16 x - 2 } { ( 8 x - 2 ) ^ { 2 } }$
C) $- \frac { 2 x } { ( 8 x - 2 ) ^ { 2 } }$
D) $- \frac { 2 } { 8 x - 2 }$

Question

Quizplus · Accepted Answer

To find the derivative of $t = \frac{x}{8x - 2}$ with respect to $x$, we use the quotient rule: $\frac{d}{dx}\left(\frac{u}{v}\right) = \frac{u'v - uv'}{v^2}$, where $u = x$ and $v = 8x - 2$. Thus, $u' = 1$ and $v' = 8$. Substituting these into the quotient rule gives $\frac{d}{dx}\left(\frac{x}{8x - 2}\right) = \frac{(1)(8x - 2) - (x)(8)}{(8x - 2)^2} = \frac{8x - 2 - 8x}{(8x - 2)^2} = -\frac{2}{(8x - 2)^2}$.

Find the Indicated Derivative dtdx\frac { \mathrm { dt } } { \mathrm { dx } }dxdt​

Find the Indicated Derivative $\frac { \mathrm { dt } } { \mathrm { dx } }$