Question

Suppose u and v are functions of x that are differentiable at x=0 and that u(0)=5, u'(0)=-5, v(0)=8, and v'(0)=-2.find the values of the following derivatives at x=0

a)d/dx(uv)=?

b)d/dx(u/v)=?

c)d/dx(v/u)=?

d)d/dx(-6v-9u)=?

Answer 1

a)-50

b)-15/32

c) 6/5

d) 57

Step-by-step explanation:

a)

`(d)/(dx)(uv)`

`u(x)v'(x)+u'(x)v(x)`

Evaluate this at
`x=0` we get:

`u(0)v'(0)+u'(0)v(0)`

`5(-2)+-5(8)`

`-10+-40`

`-50`

b)

`(d)/(dx)((u)/(v))`

`(u'(x)v(x)-u(x)v'(x))/((v(x))^2)`

Evaluate this at
`x=0` we get:

`(u'(0)v(0)-u(0)v'(0))/((v(0))^2)`

`(-5(8)-5(-2))/((8)^2)`

`\frac[-40+10}{64}`

`(-30)/(64)`

`(-15)/(32)`

c)

`(d)/(dx)((v)/(u))`

`(v'(x)u(x)-v(x)u'(x))/((u(x))^2)`

Evaluate this at
`x=0` we get:

`(v'(0)u(0)-v(0)u'(0))/((u(0))^2)`

`(-2(5)-8(-5))/((5)^2)`

`(-10+40)/(25)`

`(30)/(25)`

`(6)/(5)`

d)

`(d)/(dx)(-6v-9u)`

`-6v'(x)-9u'(x)`

Evaluate this at
`x=0`:

`-6v'(0)-9u'(0)`

`-6(-2)-9(-5)`

`12+45`

`57`