Derive (4t)/(t+1) using first principles?

Question

Please show how it was done using steps. thanks

Answer 1

First principles say:

f ' (t) = lim(h→ 0)  (f(t + h) - f(t)) / h

f ' (t) = lim(h→ 0)  (4(t + h)/(t + h + 1) - 4t / (t + 1) ) / h

= lim(h→ 0)  (4(t + h)/h(t + h + 1) - 4t / h(t + 1) )

= lim(h→ 0)  (4(t + h)(t + 1) - 4t(t + h + 1)) / (h(t + h + 1)(t + 1))

= lim(h→ 0)  4(t² + th+ t + h - t² - th - t) / (h(t + h + 1)(t + 1))

= lim(h→ 0)  4h / (h(t + h + 1)(t + 1))

= lim(h→ 0)  4 / ((t + h + 1)(t + 1))

= 4 / ((t + 1)(t + 1))

= 4 / (t + 1)²

By rule:

y = u/v

y' = (vu' - uv') / v²

y = 4t / (t + 1)

u = 4t, u ' = 4

v = (t + 1) v' = 1

y' = (vu' - uv') / v²

y ' = (4(t + 1) - 4t) / (t + 1)²

y ' = (4t + 4 - 4t) / (t + 1)²

y ' = 4 / (t + 1)²

Answer 2

I assume you mean the limit process if not please say so:

lim{h->0}

((4t+4h)/(t+h+1)-(4t)/(t+1))/h (combine fractions)->

((4t^2+4th+4t+4h-4t^2-4th-4t)/(t^2+th+... (simplify the top) ->

((4h)/(t^2+th+2t+h+1))/h (cancel h on top with /h, then sub 0 in for h)

4/(t^2+2t+1)=

4/(t+1)^2