Differentiate the function from first principle f (t)=kt^4
Let Δt be a small increment in the variable, t.
Let Δf be the corresponding increment in the function, f.
We have, to begin,
f = kt^4
After applying the increments,
f + Δf = k(t + Δt)^4
Expanding the bracketed term,
f + Δf = k(t^4 + 4t^3. Δt + 6t^2. (Δt)^2 + 4t.( Δt)^3 + + (Δt)^4)
kt^4 + Δf = k(t^4 + 4t^3. Δt + 6t^2. (Δt)^2 + 4t.( Δt)^3 + + (Δt)^4)
Δf = k(4t^3. Δt + 6t^2. (Δt)^2 + 4t.( Δt)^3 + + (Δt)^4) (cancelling out the kt^4 term)
Δf/Δt = k(4t^3 + 6t^2. (Δt) + 4t.( Δt)^2 + + (Δt)^3) (dividing both sides by Δt)
In the limit, as Δt -> zero, Δf -> zero and Δt, Δt^2, Δt^3 all -> zero and Δf/Δt -> df/dt, then
df/dt = 4kt^3