Proving Force Formula using Calculus

[arctic zenith]

Attached is the question along with my work. How do I get rid of the bracket in the numerator?

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[deft gale]

how is $\dv{t}\sqrt{1-\frac{v^2}{c^2}}=0$?

[low narwhal]

Omegabet_

[arctic zenith]

Differentiating with respect to t, c and v are constant so you're differentiating a constant and thus it's 0

[deft gale]

so then.. the entire thing would be 0 no?

cause dv/dt*sqrt()=0 by that

so 0-0/cnst = 0

v isnt a constant, c is a constant

[arctic zenith]

Yeah of course but the whole point of the question is to prove the formula so u play games with differentiation to get the two force formulas to be equal

[deft gale]

yes, but you did something bs

you said something not constant is a constant

everything that follows is wrong

[arctic zenith]

How are v and c not considered constants when differentiating in terms of t

[deft gale]

are you told v is constant?

c is the speed of light, which is famously a constant

[arctic zenith]

Doesn't matter I'm differentiating in terms of t

[deft gale]

ok

then you showed F=0

since $\dv{v}{t}=0$

[low narwhal]

Omegabet_

[deft gale]

hence the numerator is $0-0=0$

[low narwhal]

Omegabet_

[deft gale]

which means... all of physics except displacement is... constantly 0

which, idk about you, feels wrong

[arctic zenith]

Ok what do you want me to do argue with the question my answer wasn't f=0

[deft gale]

it is

your answer is 0 given you're saying v is a constant wrt t

but anyway, v isnt a constant

so treat it as such

[arctic zenith]

My brother in Christ this isn't helping me prove the formula

[deft gale]

it is

ive told you the glaring mistake you've made

and told you how to correct it

compute $\dv{t}\sqrt{1-\frac{v^2}{c^2}}$ properly

[low narwhal]

Omegabet_

[deft gale]

then continue with the algebra

[arctic zenith]

Mmm implicit differentiation?

[deft gale]

sure

but anyway, by chain rule you get $\frac{1}{2\sqrt{1-\frac{v^2}{c^2}}}\dv{t}(1-v^2/c^2)=\frac{1}{2\sqrt{1-v^2/c^2}}(-\frac{2va}{c^2})$

[low narwhal]

Omegabet_

[deft gale]

or something like that

[deft gale]

yeah no clue what this is suppose to be

$F=m_0\frac{\dv{v}{t}\sqrt{1-v^2/c^2}-v\dv{t}\sqrt{1-v^2/c^2}}{1-v^2/c^2}=m_0\frac{a\sqrt{1-v^2/c^2}+\frac{av^2}{c^2}(1-v^2/c^2)^{-1/2}}{1-v^2/c^2}$

multiplying top and bottom by $\sqrt{1-v^2/c^2}$ and further algebra should give the desired result

[low narwhal]

Omegabet_