Answered: "Find the derivative using first principles? : xn"

Briley Cabrera

Answered question

2022-04-12

Find the derivative using first principles? :

x^{n}

oanhtih6

Beginner2022-04-13Added 10 answers

Explanation:
Using the power rule:

\frac{d}{d x} (x^{n}) = n x^{n - 1}

Example:

\frac{d}{d x} (x^{4}) = (4) x^{4 - 1} = 4 x^{3}

Alonso Christian

Beginner2022-04-14Added 11 answers

Verify (by multiplication) that for positive integer n,

x^{n} - t^{n} = (x - t) (x^{n - 1} + x^{n - 2} t + x^{n - 3} t^{2} + \dots + x t^{n - 2} + t^{n - 1})

\frac{d}{d x} (x^{n}) = lim_{t \to x} \frac{x^{n} - t^{n}}{x - t}

= lim_{t \to x} \frac{(x - t) (x^{n - 1} + x^{n - 2} t + x^{n - 3} t^{2} + \dots + x t^{n - 2} + t^{n - 1})}{x - t}

= lim_{t \to x} (x^{n - 1} + x^{n - 2} t + x^{n - 3} t^{2} + \dots + x t^{n - 2} + t^{n - 1})

There are n terms, each with limit

x^{n - 1}

, so the limit is

\frac{d}{d x} (x^{n}) = n x^{n - 1}

for positive integer n.

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