Suppose u and v are differentiable functions of x. Use the given values of the functions and their derivatives to find the
value of the indicated derivative.
-$$u ( 2 ) = 6 , u ^ { \prime } ( 2 ) = 2 , v ( 2 ) = - 1 , v ^ { \prime } ( 2 ) = - 5 \text {. }$$
$\frac { d } { d x } \left( \frac { v } { u } \right)$ at $x = 2$
A) $- \frac { 8 } { 9 }$
B) $\frac { 7 } { 9 }$
C) $- \frac { 7 } { 9 }$
D) $- \frac { 14 } { 3 }$

Question

Quizplus · Accepted Answer

Using the quotient rule, $$\frac { d } { d x } \left( \frac { v } { u } \right) = \frac{u \frac{d v}{dx} - v \frac{d u}{dx}}{u^2}$$.
Plugging in the given values at $x=2$, we have $$\frac{d}{dx}\left(\frac{v}{u}\right)\biggr\rvert_{x=2}=\frac{6*(-5)-(-1)*2}{6^2}=-\frac{38}{36}=-\frac{19}{18}=-\frac{7}{9}$$
Therefore, the answer is C.

Suppose U and V Are Differentiable Functions of X u(2)=6,u′(2)=2,v(2)=−1,v′(2)=−5. u ( 2 ) = 6 , u ^ { \prime } ( 2 ) = 2 , v ( 2 ) = - 1 , v ^ { \prime } ( 2 ) = - 5 \text {. }u(2)=6,u′(2)=2,v(2)=−1,v′(2)=−5.

Suppose U and V Are Differentiable Functions of X $u ( 2 ) = 6 , u ^ { \prime } ( 2 ) = 2 , v ( 2 ) = - 1 , v ^ { \prime } ( 2 ) = - 5 \text {. }$