Question

I'm trying to calculate the derivative of this: P= mv/ sqrt(1-v^2/c^2)where m and c are constants, and v depends on t. I know about the chain and product rules but I'm not even sure where to start. :/ Thank you for your help.

Answer 1

p = mv * (1-v^2/c^2)^(-1/2)
p' = m * (1-v^2/c^2)^(-1/2) - mv * (-1/2) (1-v^2/c^2)^(-3/2) * (-2v/c^2)

= [m(1-v^2/c^2) + mv^2/c^2]/(1-v^2/c^2)^(-3/2)
= m/(1-v^2/c^2)&(3/2)

Answer 2

Can you please explain how you derived the first half of the second line before the '-mv * (...)'? Thank you.

Answer 3

product rule

p = f*g
p' = f'g + fg'

Answer 4

I'm still not quite sure how that works out, if you can explain it please? Isn't the derivative of v dv/dt and the derivative of m (since it's a constant) 0?

Answer 5

f = mv
f' = m

g = (1-v^2/c^2)^(-1/2)
using the chain rule, g = h^(-1/2), so g' = (-1/2) h^(-3/2) h'

g' = (-1/2) (1-v^2/c^2)^(-3/2) (-2v/c^2)

and the rest follows