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.
-$$\mathrm { u } ( 2 ) = 7 , \mathrm { u } ^ { \prime } ( 2 ) = 4 , \mathrm { v } ( 2 ) = - 3 , \mathrm { v } ^ { \prime } ( 2 ) = - 4 \text {. }$$
$\frac { \mathrm { d } } { \mathrm { dx } } \left( \frac { \mathrm { u } } { \mathrm { v } } \right)$ at $\mathrm { x } = 2$
A) $- \frac { 16 } { 3 }$
B) $- \frac { 40 } { 9 }$
C) 1
D) $\frac { 16 } { 9 }$

Question

Quizplus · Accepted Answer

Using the quotient rule, we have

$$\frac { d } { d x } \left( \frac { u } { v } \right) = \frac { v \frac { d u } { d x } - u \frac { d v } { d x } } { v ^ { 2 } }$$

At x = 2, we have u(2) = 7, u'(2) = 4, v(2) = -3, and v'(2) = -4. Substituting these values in the formula above, we get

$$\frac { d } { d x } \left( \frac { u } { v } \right) \bigg|_{x=2} = \frac {-3(4) - 7(-4)} { (-3)^2 } = \frac { 16 } { 9 } $$

Therefore, the answer is D.

Suppose U and V Are Differentiable Functions of X u(2)=7,u′(2)=4,v(2)=−3,v′(2)=−4. \mathrm { u } ( 2 ) = 7 , \mathrm { u } ^ { \prime } ( 2 ) = 4 , \mathrm { v } ( 2 ) = - 3 , \mathrm { v } ^ { \prime } ( 2 ) = - 4 \text {. }u(2)=7,u′(2)=4,v(2)=−3,v′(2)=−4.

Suppose U and V Are Differentiable Functions of X $\mathrm { u } ( 2 ) = 7 , \mathrm { u } ^ { \prime } ( 2 ) = 4 , \mathrm { v } ( 2 ) = - 3 , \mathrm { v } ^ { \prime } ( 2 ) = - 4 \text {. }$