Derivative: Difference between revisions
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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>. | 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>. | ||
== Criticism == | |||
This definition makes things way more complicated. I will demonstrate this with the following example. Let's suppose that <math>f(x) = x^3</math>, so <math>(\Delta_\epsilon f)(x) = 3x^2 + 3\epsilon x + \epsilon^2 </math> and <math>(\Delta_\epsilon f - \Delta_{\epsilon'} f)(x) = 3 (\epsilon - \epsilon' )x + ( \epsilon^3 - \epsilon'^3)</math>. And let's say that we are in a context where anything with absolute value below 1.0 is nill. Let <math>\epsilon = 0.91 </math> and <math>\epsilon' = 0.01</math>. Then <math>(\Delta_\epsilon f - \Delta_{\epsilon'} f)(x) = 2.7 x + 0.75357</math>. This is greater than 1 when <math>x > 0.0912\overline{703}</math>, so we reach the seemingly absurd conclusion that <math>f(x) = x^3</math> is not differentiable where <math>x > 0.0912\overline{703}</math>. | |||
This problem is not present in the standard definition of derivative, where the remainder terms go away in the limit and we are just left with <math>f'(x) = 3 x^2 </math>. | |||
Revision as of 21:55, 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 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 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}} , 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}} .
Criticism
This definition makes things way more complicated. I will demonstrate this with the following example. Let's suppose 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(x) = x^3} , so 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) = 3x^2 + 3\epsilon x + \epsilon^2 } and 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 - \Delta_{\epsilon'} f)(x) = 3 (\epsilon - \epsilon' )x + ( \epsilon^3 - \epsilon'^3)} . And let's say that we are in a context where anything with absolute value below 1.0 is nill. 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 = 0.91 } and 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' = 0.01} . Then 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 - \Delta_{\epsilon'} f)(x) = 2.7 x + 0.75357} . This is greater than 1 when 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 > 0.0912\overline{703}} , so we reach the seemingly absurd conclusion that is not differentiable where 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 > 0.0912\overline{703}} .
This problem is not present in the standard definition of derivative, where the remainder terms go away in the limit and we are just left with 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) = 3 x^2 } .