Question

Find derivative of f(x) = 4x³ using first Principre and F(x) = 2x²-3x ​

Answer 1

12x^2.

Explanation:

let y = 4x^3

If y is increased by a small amount Δy then x is increased by a small amount Δx, so we write:

y + Δy = 4(x + Δx)^3

y + Δy = 4(x^3 + 3x^2 Δx + 3x(Δx)^2 + (Δx)^3)

y + Δy = 4x^3 + 12x^2 Δx + 12x(Δx)^2 + 4(Δx)^3

Δy = 4x^3 + 12x^2 Δx + 12x(Δx)^2 + 4(Δx)^3 - y

= 4x^3 + 12x^2 Δx + 12x(Δx)^3 + 4(Δx)^2 - 4x^3

Δy/Δx = (4x^3 + 12x^2 Δx + 12x(Δx)^3 + 4(Δx)^2- 4x^3 ) / Δx

= (12x^2 Δx + 12x(Δx)^3 + 4(Δx)^2) / Δx

= 12x^2 + 12x(Δx)^2 + 4Δx

Now as Δx approaches zero, the limit ( that is the derivative) of f(x)

= 12x^2

- as we can neglect the last 2 terms.

The second part can be solved by the same process.