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Supposing that s' is the new position, s the old one, t the time started, t' the new time, Δt the time difference, (approaches zero) and v the velocity, or the derivative of the position with respect to time: s' = s + vt. So far so good, I'll work out the derivative first:

ds/dt = vt + vΔt - vt/vΔt

= v.

That shows that the derivative of ds/dt is v. However, most calculus equations are given as s = ƒ(t) which can be something like s = 4t^2 + 2t + 20. How is this possible? I know that the equation of a straight line can be written as : y = mx + c which in this case would be s' = vt + s. So how is it possible, without knowing the velocity, to find an equation such as the s = ƒ(t)? one here? Besides doesn't this equation hold only for a straight line? So would it apply throughout an entire graph if the velocity, that is the slope changes? And if so, why?

Any help would be greatly appreciated, thanks

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# Using calculus to find velocity

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