Mo logo [home] [lexicon] [problems] [tests] [courses] [auxiliaries] [notes] [staff] german flag
Mathematik-Online lexicon:

Derivative of f(x)=x^2

A B C D E F G H I J K L M N O P Q R S T U V W X Y Z overview
According to the definition, the derivative of the function $ f(x)=x^2$ is given by

$\displaystyle f^\prime(x) = \lim_{h\to0} \frac{(x+h)^2-x^2}{h}=\lim_{h\to0} \frac{2xh+h^2}{h}=\lim_{h\to0} (2x+h)=2x\,. $

The seconed derivative is constant: $ f^{\prime\prime}(x) = 2$.

In general, with the aid of the binomial formula, we can compute the derivative of an arbitrary monomial $ f(x) = x^n$:

$\displaystyle f^\prime(x) = \lim_{h \to0} \frac{(x+h)^n - x^n}{h} = \lim_{h \to0} \frac{\binom{n}{1} x^{n-1}h + O(h^2)}{h} = nx^{n-1}\,, $

with $ O(h^2)$ denoting terms of order $ h^2$.
[Links]
  automatisch erstellt am 14.  6. 2016