Question

Find the differential of `f(x)=x^20` using first principal?

Answer 1

`(df)/(dx)=20x^19`

Explanation:

Let `f(x)=x^20`

then using bimonial expansion `f(x+h)=(x+h)^20=x^20+20x^19h+(20*19)/(1*2)x^18h^2+(20*19*18)/(1*2*3)x^17h^3+...`

= `x^20+20x^19h+190x^18h^2+1140x^17h^3+...`

Hence, `(df)/(dx)=Lim_(h->0)(f(x+h)-f(x))/h`

= `Lim_(h->0)(x^20+20x^19h+190x^18h^2+1140x^17h^3+...-x^20)/h`

= `Lim_(h->0)(20x^19+190x^18h+1140x^17h^2+...)`

= `20x^19`

Answer 2

See below.

Explanation:

`f'(x) = lim_(trarrx)(t^20-x^20)/(t-x)`

`= lim_(trarrx)(t^19+t^18x+t^17x^2+ * * * + t^2x^17+tx^18+t^19)`

`= underbrace(x^19+x^18x+x^17x^2+ * * * x^2x^17+x x^18+x^19)_(20 " terms, each is "x^19)`

`= 20x^19`