Differenciate the function from first principle f (t)=kt^

Question

F (t)=kt^4 differentiate using first principle

Answer 1

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