find the derivative using limit laws for f((t^-1)-t)

Question

how to find the derivative using limit laws

Answer 1

Do you mean f(t)=1/t-t?

f(t+k)=1/(t+k)-t-k where k is a small increment.

The rate of change is the derivative=(f(t+k)-f(t))/k in the limit as k→0.

f(t)=(1-t²)/t, f(t+k)=(1-(t+k)²)/(t+k).

Δf=f(t+k)-f(t)=(1-(t+k)²)/(t+k)-(1-t²)/t,

Δf=[t(1-(t+k)²)-(t+k)(1-t²)]/[t(t+k)],

Δf=[t-t³-2kt²-tk²-t-k+t³+kt²]/(t²+kt),

Δf=(-kt²-tk²-k)/(t²+kt).

As k→0, tk²→0 and kt is very small compared to t², so:

Δf→(-kt²-k)/t²=-k-k/t², and Δf/k→df/dt=-1-1/t².