Derivative: Difference between revisions
(Created page with "Let <math>f : \mathbb{Q} \rightarrow \mathbb{Q}</math>, and let <math>\epsilon : \mathbb{Q}</math> be positive. We define <math>\Delta_\epsilon f : \mathbb{Q} \rightarrow \mathbb{Q}</math> by<math display="block">(\Delta_\epsilon f) (x) := \frac{f(x + \epsilon) - f(x)}{\epsilon}. </math>We say that <math>f</math> is '''differentiable''' if is .... nill whenever") |
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Let <math>f : \mathbb{Q} \rightarrow \mathbb{Q}</math>, and let <math>\epsilon : \mathbb{Q}</math> | Let <math>f : \mathbb{Q} \rightarrow \mathbb{Q}</math>, and let <math>\epsilon : \mathbb{Q}_{>0}</math>. I define <math>\Delta_\epsilon f : \mathbb{Q} \rightarrow \mathbb{Q}</math> by<math display="block">(\Delta_\epsilon f) (x) := \frac{f(x + \epsilon) - f(x)}{\epsilon}, \quad x : \mathbb{Q}. </math>I say that <math>f : \mathbb{Q} \rightarrow \mathbb{Q}</math> is '''differentiable''' at <math>x:\mathbb{Q}</math>, if <math>(\Delta_\epsilon f)(x) - (\Delta_{\epsilon'} f)(x) </math> is [[nill]] whenever <math>\epsilon, \epsilon' : \mathbb{Q}_{>0} </math> are both nill. | ||
If <math>f : \mathbb{Q} \rightarrow \mathbb{Q}</math> is differentiable at <math>x:\mathbb{Q}</math>, then I define the '''derivative''' of <math>f</math> at <math>x</math> as <math>f'(x) := (\Delta_\epsilon f)(x) </math>, for some nill <math>\epsilon : \mathbb{Q}_{>0}</math>. | |||
Revision as of 20:25, 26 January 2024
Let Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f : \mathbb{Q} \rightarrow \mathbb{Q}} , and let Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \epsilon : \mathbb{Q}_{>0}} . I define Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Delta_\epsilon f : \mathbb{Q} \rightarrow \mathbb{Q}} byFailed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (\Delta_\epsilon f) (x) := \frac{f(x + \epsilon) - f(x)}{\epsilon}, \quad x : \mathbb{Q}. } I say that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f : \mathbb{Q} \rightarrow \mathbb{Q}} is differentiable at Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x:\mathbb{Q}} , if Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (\Delta_\epsilon f)(x) - (\Delta_{\epsilon'} f)(x) } is nill whenever Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \epsilon, \epsilon' : \mathbb{Q}_{>0} } are both nill.
If Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f : \mathbb{Q} \rightarrow \mathbb{Q}} is differentiable at , then I define the derivative of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f} at Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} as Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f'(x) := (\Delta_\epsilon f)(x) } , for some nill Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \epsilon : \mathbb{Q}_{>0}} .
