25-44 Differentiate. 28. J(v)=\left(v^(3)-2 v\right)\left(v^(-4)+v^(-2)\right)

Question

25-44 Differentiate.
28.
`J(v)=\left(v^(3)-2 v\right)\left(v^(-4)+v^(-2)\right)`

Answer 1

Make things easier by expanding J(v) first:

`J(v) = (v^3-2v) \left(\frac1{v^4}+\frac1{v^2}\right) = \frac1v - \frac2{v^3} - \frac2v + v = v - \frac1v - \frac2{v^3}`

Now differentiate term-by-term (power rule):

`J'(v) = \boxed{1 - \frac1{v^2} + \frac6{v^4}}`

In case you're supposed to use the product rule first, we have

`J'(v) = (3v^2 - 2) \left(\frac1{v^4} + \frac1{v^2}\right) + (v^3-2v) \left(-\frac4{v^5} - \frac2{v^3}\right)`

Expanding and simplifying will yield the same result as before.