Question

Suppose u and v are functions of x that are differentiable at x=0 and that u(0)=5 u'(0)-3 v(0)=-1 v'(0)=2 find the values of following derivatives atx=0:
a) d/dx(uv)

Answer 1

Answer:

Step-by-step explanation:

d/dx(uv) = uv'+vu'

=5*2+(-1) (-3)

=10+3=13

Answer 2

The correct question: Suppose u and v are functions of x that are differentiable at x = 0 and that u(0) = 5, u'(0) = -3,  v(0) = -1, and v'(0) = 2. Find the values of the following derivatives at x = 0.

a). $\frac{d}{dx} (uv)$

b). $\frac{d}{dx} (\frac{u}{v} )$

c). $\frac{d}{dx} (\frac{v}{u} )$

d). $\frac{d}{dx} (7v-2u)$.

The correct answers are as follows.

Given:

Given that, u and v are functions of x that are differentiable at x = 0.

Also, at x = 0,

u(0) = 5, u'(0) = -3, v(0) = -1, and v'(0) = 2.

To Find:

We have to find the values of the given derivatives at x = 0.

Solution:

Consider the first option a). $\frac{d}{dx} (uv)$.

$\frac{d}{dx} (uv) = u.v' +u'.v$

Substituting the given values in the above equation, we get,

$\frac{d}{dx} (uv)|_{x=0} = u(0).v'(0) +u'(0).v(0)$

$i.e., \frac{d}{dx} (uv)x|_{x=0} = 5$ × $2 + (-3)$ × $(-1) = 13.$

Consider the second option b). $\frac{d}{dx} (\frac{u}{v} )$.

$\frac{d}{dx} (\frac{u}{v}) = \frac{v.u' -v'.u}{v^2}$

Substituting the given values in the above equation, we get,

$\frac{d}{dx} (\frac{u}{v})|_{x=0} = \frac{v(0).u'(0) -v'(0).u(0)}{v(0)^2}$

$i.e., \frac{d}{dx} (\frac{u}{v})|_{x=0} = \frac{(-1). (-3)- (2).(5)}{(-1)^2} = -7.$

Consider the third option c). $\frac{d}{dx} (\frac{v}{u} )$.

$\frac{d}{dx} (\frac{v}{u} ) = \frac{u.v' -u'.v}{u^2}$

Substituting the given values in the above equation, we get,

$\frac{d}{dx} (\frac{v}{u})|_{x=0} = \frac{u(0).v'(0) -u'(0).v(0)}{u(0)^2}$

$i.e., \frac{d}{dx} (\frac{v}{u})|_{x=0} = \frac{(5). (2)- (-3).(-1)}{(5)^2} = \frac{7}{25}.$

Consider the fourth option d). $\frac{d}{dx} (7v-2u)$.

$\frac{d}{dx} (7v-2u) = (7v'-2u')$

Substituting the given values in the above equation, we get,

$\frac{d}{dx} (7v-2u)|_{x=0} = 7v'(0)-2u'(0)$

$i.e., \frac{d}{dx} (7v-2u)|_{x=0} = 7$ × $2 - 2$ × $(-3) = 20.$

Hence, the values of the given derivatives at x = 0 are 13, -7, $\frac{7}{25}$, and 20.

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