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.
-$$u ( 1 ) = 2 , u u ^ { \prime } ( 1 ) = - 5 , v ( 1 ) = 7 , v ^ { \prime } ( 1 ) = - 3 \text {. }$$
$\frac { d } { d x } \left( \frac { v } { u } \right)$ at $x = 1$
A) $- \frac { 29 } { 4 }$
B) $\frac { 29 } { 4 }$
C) $- \frac { 41 } { 4 }$
D) $\frac { 29 } { 2 }$

Question

Quizplus · Accepted Answer

Using the quotient rule, we have
$$\frac{d}{dx}\left(\frac{v}{u}\right) = \frac{u\frac{d}{dx}(v) - v\frac{d}{dx}(u)}{u^2} = \frac{u v' - v u'}{u^2}.$$
Plugging in the given values, we get
$$\frac{d}{dx}\left(\frac{v}{u}\right)\bigg|_{x=1} = \frac{u(1)v'(1) - v(1)u'(1)}{u(1)^2} = \frac{2(-3) - 7(-5)}{2^2} = \frac{29}{4}.$$

Suppose U and V Are Differentiable Functions of X u(1)=2,uu′(1)=−5,v(1)=7,v′(1)=−3. u ( 1 ) = 2 , u u ^ { \prime } ( 1 ) = - 5 , v ( 1 ) = 7 , v ^ { \prime } ( 1 ) = - 3 \text {. }u(1)=2,uu′(1)=−5,v(1)=7,v′(1)=−3.

Suppose U and V Are Differentiable Functions of X $u ( 1 ) = 2 , u u ^ { \prime } ( 1 ) = - 5 , v ( 1 ) = 7 , v ^ { \prime } ( 1 ) = - 3 \text {. }$