Difference between revisions of "Derivative"
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Let $X$ be a subset of [[real number|real numbers]], $x_0 \in X$, and let $f \colon X \rightarrow \mathbb{R}$ be a function. We say that $f$ is differentiable at $x_0$ if the [[limit]] | Let $X$ be a subset of [[real number|real numbers]], $x_0 \in X$, and let $f \colon X \rightarrow \mathbb{R}$ be a function. We say that $f$ is differentiable at $x_0$ if the [[limit]] | ||
$$f'(x_0)=\displaystyle\lim_{h \rightarrow 0} \dfrac{f(x_0+h)-f(x_0)}{h}$$ | $$f'(x_0)=\displaystyle\lim_{h \rightarrow 0} \dfrac{f(x_0+h)-f(x_0)}{h}$$ | ||
Revision as of 05:03, 16 September 2016
Let $X$ be a subset of real numbers, $x_0 \in X$, and let $f \colon X \rightarrow \mathbb{R}$ be a function. We say that $f$ is differentiable at $x_0$ if the limit $$f'(x_0)=\displaystyle\lim_{h \rightarrow 0} \dfrac{f(x_0+h)-f(x_0)}{h}$$ exists.
Properties
Theorem
The derivative operator is a linear operator.
Proof
References
Theorem
The following formula holds: $$\displaystyle\lim_{q \rightarrow 1^+} D_q f(x) = f'(x),$$ where $D_q$ denotes the $q$-derivative and $f'(x)$ denotes the derivative of $f$.
