Derivative: Difference between revisions
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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. | Let <math>f : \mathbb{Q} \rightarrow \mathbb{Q}</math> be a [[function]], 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>. | 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 22:17, 28 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}} be a function, 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 , if is nill whenever are both nill.
If is differentiable at , then I define the derivative of at as , for some nill .
Criticism
This definition makes things way more complicated. I will demonstrate this with the following example. Let's suppose that 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^2 - \epsilon'^2)} . And let's say that we are in a context where anything with absolute value below 0.1 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 = 0.091 } 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' = 0.001} . 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) = 0.27 x + 0.00828 } . This quantity is greater than 0.1, and thus non-nill, 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 > 0.339\overline{703}} , so we reach the seemingly absurd conclusion 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} 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 > 0.339\overline{703}} .
This complication 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 } . It's not completely clear whether or not this complication is a problem. Those pesky terms are measuring something real, which calculus is ignoring. They are measuring the difference between two different methods of finding the slope of the tangent line to a real curve.