Derivative - Revision history

Vipul: /* Algebraic definition elaborated in terms of epsilon-delta definition of limits */

2024-04-11T02:39:34Z

Algebraic definition elaborated in terms of epsilon-delta definition of limits

← Older revision		Revision as of 02:39, 11 April 2024
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	===Algebraic definition elaborated in terms of epsilon-delta definition of limits===		===Algebraic definition elaborated in terms of epsilon-delta definition of limits===

	Suppose <math>f</math> is a [[function]] defined on a subset of the reals and <math>x_0</math> is a point in the [[interior]] of the [[domain]] of <math>f</math>, i.e., the domain of <math>f</math> contains an [[open interval]] surrounding <math>x_0</math>. The '''derivative''' (also called '''first derivative''') of <math>f</math> at <math>x_0</math>, denoted <math>\! f'(x_0)</math>, is defined as a real number <math>L</math> such that:		Suppose <math>f</math> is a [[function]] defined on a subset of the reals and <math>x_0</math> is a point in the [[interior]] of the [[domain]] of <math>f</math>, i.e., the domain of <math>f</math> contains an [[open interval]] surrounding <math>x_0</math>. The '''derivative''' (also called '''first derivative''') of <math>f</math> at <math>x_0</math>, denoted <math>f'(x_0)</math>, is defined as a real number <math>L</math> such that:

	{{quotation\|For every <math>\varepsilon > 0</math><br> there exists <math>\delta > 0</math> such that <br>if <math>\! 0 < \|x - x_0\| < \delta</math><br> then <math>\|f(x) - f(x_0) - L(x - x_0)\| < \varepsilon\|x - x_0\|</math>.}}		{{quotation\|For every <math>\varepsilon > 0</math><br> there exists <math>\delta > 0</math> such that <br>if <math>\! 0 < \|x - x_0\| < \delta</math><br> then <math>\|f(x) - f(x_0) - L(x - x_0)\| < \varepsilon\|x - x_0\|</math>.}}

Vipul: /* Algebraic definition elaborated in terms of epsilon-delta definition of limits */

2024-04-11T02:39:19Z

Algebraic definition elaborated in terms of epsilon-delta definition of limits

← Older revision		Revision as of 02:39, 11 April 2024
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	Suppose <math>f</math> is a [[function]] defined on a subset of the reals and <math>x_0</math> is a point in the [[interior]] of the [[domain]] of <math>f</math>, i.e., the domain of <math>f</math> contains an [[open interval]] surrounding <math>x_0</math>. The '''derivative''' (also called '''first derivative''') of <math>f</math> at <math>x_0</math>, denoted <math>\! f'(x_0)</math>, is defined as a real number <math>L</math> such that:		Suppose <math>f</math> is a [[function]] defined on a subset of the reals and <math>x_0</math> is a point in the [[interior]] of the [[domain]] of <math>f</math>, i.e., the domain of <math>f</math> contains an [[open interval]] surrounding <math>x_0</math>. The '''derivative''' (also called '''first derivative''') of <math>f</math> at <math>x_0</math>, denoted <math>\! f'(x_0)</math>, is defined as a real number <math>L</math> such that:

	{{quotation\|For every <math>\!\varepsilon > 0</math><br> there exists <math>\!\delta > 0</math> such that <br>if <math>\! 0 < \|x - x_0\| < \delta</math><br> then <math>\! \|f(x) - f(x_0) - L(x - x_0)\| < \varepsilon\|x - x_0\|</math>.}}		{{quotation\|For every <math>\varepsilon > 0</math><br> there exists <math>\delta > 0</math> such that <br>if <math>\! 0 < \|x - x_0\| < \delta</math><br> then <math>\|f(x) - f(x_0) - L(x - x_0)\| < \varepsilon\|x - x_0\|</math>.}}

	<center>{{#widget:YouTube\|id=cZVLsvvBDgo}}</center>		<center>{{#widget:YouTube\|id=cZVLsvvBDgo}}</center>

Vipul: /* Definition as a function */

2024-04-11T02:36:15Z

Definition as a function

← Older revision		Revision as of 02:36, 11 April 2024
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	==Definition as a function==		==Definition as a function==

	Suppose <math>f</math> is a [[function]] defined on a subset of the reals. Its '''derivative''' or '''first derivative''', denoted <math>\! f'</math>, is a function defined as follows:		Suppose <math>f</math> is a [[function]] defined on a subset of the reals. Its '''derivative''' or '''first derivative''', denoted <math>f'</math>, is a function defined as follows:

	* The domain is the following subset of the domain of <math>\! f</math>: An element in the domain of <math>\! f</math> is in the domain of <math>\! f'</math> if and only if it is in the ''interior'' of the domain of <math>\! f</math> and the derivative of <math>\! f</math> exists at the point.		* The domain is the following subset of the domain of <math>f</math>: An element in the domain of <math>f</math> is in the domain of <math>f'</math> if and only if it is in the ''interior'' of the domain of <math>f</math> and the derivative of <math>f</math> exists at the point.
	* The function value at any point in the domain is simply the value of the derivative of <math>\! f</math> at that point.		* The function value at any point in the domain is simply the value of the derivative of <math>f</math> at that point.

	{{generic point specific point confusion}}		{{generic point specific point confusion}}

Vipul: /* Right hand derivative */

2021-09-25T03:17:35Z

Right hand derivative

← Older revision		Revision as of 03:17, 25 September 2021
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	<math>\operatorname{LHD}(f)(x_0) = f'_-(x_0) := \lim_{x \to x_0^-}\Delta f(x,x_0) = \lim_{x \to x_0^-} \frac{f(x) - f(x_0)}{x - x_0} = \lim_{h \to 0^-} \frac{f(x_0 + h) - f(x_0))}{h}</math>		<math>\operatorname{LHD}(f)(x_0) = f'_-(x_0) := \lim_{x \to x_0^-}\Delta f(x,x_0) = \lim_{x \to x_0^-} \frac{f(x) - f(x_0)}{x - x_0} = \lim_{h \to 0^-} \frac{f(x_0 + h) - f(x_0))}{h}</math>

	===Right hand derivative===		===Right-hand derivative===

	Suppose <math>f</math> is a function defined at a point <math>x_0 \in \R</math> and also to the immediate right of <math>x_0</math>. The '''right hand derivative''' of <math>f</math> at <math>x_0</math> is defined as the ''right hand limit'' for the difference quotient between <math>x</math> and <math>x_0</math>. In other words, it is:		Suppose <math>f</math> is a function defined at a point <math>x_0 \in \R</math> and also to the immediate right of <math>x_0</math>. The '''right-hand derivative''' of <math>f</math> at <math>x_0</math> is defined as the ''right hand limit'' for the difference quotient between <math>x</math> and <math>x_0</math>. In other words, it is:

	<math>\operatorname{RHD}(f)(x_0) = f'_+(x_0) := \lim_{x \to x_0^+} \Delta f(x,x_0) = \lim_{x \to x_0^+} \frac{f(x) - f(x_0)}{x - x_0} = \lim_{h \to 0^+} \frac{f(x_0 + h) - f(x_0)}{h}</math>		<math>\operatorname{RHD}(f)(x_0) = f'_+(x_0) := \lim_{x \to x_0^+} \Delta f(x,x_0) = \lim_{x \to x_0^+} \frac{f(x) - f(x_0)}{x - x_0} = \lim_{h \to 0^+} \frac{f(x_0 + h) - f(x_0)}{h}</math>

Vipul: /* Left hand derivative */

2021-09-25T03:17:28Z

Left hand derivative

← Older revision		Revision as of 03:17, 25 September 2021
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	==One-sided notions==		==One-sided notions==

	===Left hand derivative===		===Left-hand derivative===

	Suppose <math>f</math> is a function defined at a point <math>x_0 \in \R</math> and also to the immediate left of <math>x_0</math>. The '''left hand derivative''' of <math>f</math> at <math>x_0</math> is defined as the ''left hand limit'' for the difference quotient between <math>x</math> and <math>x_0</math>. In other words, it is:		Suppose <math>f</math> is a function defined at a point <math>x_0 \in \R</math> and also to the immediate left of <math>x_0</math>. The '''left-hand derivative''' of <math>f</math> at <math>x_0</math> is defined as the ''left hand limit'' for the difference quotient between <math>x</math> and <math>x_0</math>. In other words, it is:

	<math>\operatorname{LHD}(f)(x_0) = f'_-(x_0) := \lim_{x \to x_0^-}\Delta f(x,x_0) = \lim_{x \to x_0^-} \frac{f(x) - f(x_0)}{x - x_0} = \lim_{h \to 0^-} \frac{f(x_0 + h) - f(x_0))}{h}</math>		<math>\operatorname{LHD}(f)(x_0) = f'_-(x_0) := \lim_{x \to x_0^-}\Delta f(x,x_0) = \lim_{x \to x_0^-} \frac{f(x) - f(x_0)}{x - x_0} = \lim_{h \to 0^-} \frac{f(x_0 + h) - f(x_0))}{h}</math>

Vipul: /* Algebraic definition elaborated in terms of epsilon-delta definition of limits */

2021-09-25T01:46:00Z

Algebraic definition elaborated in terms of epsilon-delta definition of limits

← Older revision		Revision as of 01:46, 25 September 2021
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	Suppose <math>f</math> is a [[function]] defined on a subset of the reals and <math>x_0</math> is a point in the [[interior]] of the [[domain]] of <math>f</math>, i.e., the domain of <math>f</math> contains an [[open interval]] surrounding <math>x_0</math>. The '''derivative''' (also called '''first derivative''') of <math>f</math> at <math>x_0</math>, denoted <math>\! f'(x_0)</math>, is defined as a real number <math>L</math> such that:		Suppose <math>f</math> is a [[function]] defined on a subset of the reals and <math>x_0</math> is a point in the [[interior]] of the [[domain]] of <math>f</math>, i.e., the domain of <math>f</math> contains an [[open interval]] surrounding <math>x_0</math>. The '''derivative''' (also called '''first derivative''') of <math>f</math> at <math>x_0</math>, denoted <math>\! f'(x_0)</math>, is defined as a real number <math>L</math> such that:

	{{quotation\|For every <math>\!\varepsilon > 0</math><br> there exists <math>\!\delta > 0</math> such that <br>if <math>\! 0 < \|x - x_0\| < \delta</math><br> then <math>\! \|f(x) - f(x_0) - L(x - x_0)\| < \~~epsilon~~\|x - x_0\|</math>.}}		{{quotation\|For every <math>\!\varepsilon > 0</math><br> there exists <math>\!\delta > 0</math> such that <br>if <math>\! 0 < \|x - x_0\| < \delta</math><br> then <math>\! \|f(x) - f(x_0) - L(x - x_0)\| < \varepsilon\|x - x_0\|</math>.}}

	<center>{{#widget:YouTube\|id=cZVLsvvBDgo}}</center>		<center>{{#widget:YouTube\|id=cZVLsvvBDgo}}</center>

Vipul: /* Algebraic definition elaborated in terms of epsilon-delta definition of limits */

2021-09-25T01:45:53Z

Algebraic definition elaborated in terms of epsilon-delta definition of limits

← Older revision		Revision as of 01:45, 25 September 2021
Line 50:		Line 50:
	Suppose <math>f</math> is a [[function]] defined on a subset of the reals and <math>x_0</math> is a point in the [[interior]] of the [[domain]] of <math>f</math>, i.e., the domain of <math>f</math> contains an [[open interval]] surrounding <math>x_0</math>. The '''derivative''' (also called '''first derivative''') of <math>f</math> at <math>x_0</math>, denoted <math>\! f'(x_0)</math>, is defined as a real number <math>L</math> such that:		Suppose <math>f</math> is a [[function]] defined on a subset of the reals and <math>x_0</math> is a point in the [[interior]] of the [[domain]] of <math>f</math>, i.e., the domain of <math>f</math> contains an [[open interval]] surrounding <math>x_0</math>. The '''derivative''' (also called '''first derivative''') of <math>f</math> at <math>x_0</math>, denoted <math>\! f'(x_0)</math>, is defined as a real number <math>L</math> such that:

	{{quotation\|For every <math>\!\~~epsilon~~ > 0</math><br> there exists <math>\!\delta > 0</math> such that <br>if <math>\! 0 < \|x - x_0\| < \delta</math><br> then <math>\! \|f(x) - f(x_0) - L(x - x_0)\| < \epsilon\|x - x_0\|</math>.}}		{{quotation\|For every <math>\!\varepsilon > 0</math><br> there exists <math>\!\delta > 0</math> such that <br>if <math>\! 0 < \|x - x_0\| < \delta</math><br> then <math>\! \|f(x) - f(x_0) - L(x - x_0)\| < \epsilon\|x - x_0\|</math>.}}

	<center>{{#widget:YouTube\|id=cZVLsvvBDgo}}</center>		<center>{{#widget:YouTube\|id=cZVLsvvBDgo}}</center>

Vipul: /* Computation of derivative */

2021-09-24T14:48:52Z

Computation of derivative

← Older revision		Revision as of 14:48, 24 September 2021
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	==Computation of derivative==		==Computation of derivative==

			=== List of most commonly used rules ===

	<section begin="differentiation rules"/>		<section begin="differentiation rules"/>
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	<section end="differentiation rules"/>		<section end="differentiation rules"/>

			=== Significance of differentiation rules ===

			The list of differentiation rules is fairly complete with respect to the most typical ways of constructing functions from other functions. This means that if we have explicit expressions for the derivatives of a collection of functions, we can obtain explicit expressions for the derivatives of any other function constructed from them via any of the methods covered in the table above (pointwise sum, pointwise difference, scalar multiple, pointwise product, pointwise quotient, composite, inverse, and piecewise definition).

			Computer programs that implement symbolic mathematics (such as Mathematica) generally have all these rules coded in, and are able to use them to differentiate any function constructed by applying these operations to functions whose derivative is already known by the program.

Vipul: /* Computation of derivative */

2021-09-24T14:25:07Z

Computation of derivative

← Older revision		Revision as of 14:25, 24 September 2021
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	\| [[inverse function]] of a [[one-one function]] \|\| <math>f^{-1}</math> sends <matH>x</math> to the unique <math>y</math> such that <math>f(y) = x</math> \|\| <math>\! \frac{1}{f'(f^{-1}(x))}</math> \|\| [[inverse function theorem]]		\| [[inverse function]] of a [[one-one function]] \|\| <math>f^{-1}</math> sends <matH>x</math> to the unique <math>y</math> such that <math>f(y) = x</math> \|\| <math>\! \frac{1}{f'(f^{-1}(x))}</math> \|\| [[inverse function theorem]]
	\|-		\|-
	\| [[piecewise definition of functions\|piecewise definition]] \|\| {{~~fillin~~}} \|\| ~~{{fillin}}~~ \|\| [[differentiation rule for piecewise definition by interval]]		\| [[piecewise definition of functions\|piecewise definition]] \|\| <math>f(x) := \left\lbrace \begin{array}{rl} f_1(x), & x < c \\ f_2(x), & c < x \\v, & x = c \end{array}\right.</math> where <math>f_1, f_2</math> can be extended to differentiable functions on all reals\|\| <math>f' = f_1'</math> to the left of <math>c</math> and <math>f' = f_2'</math> to the right of <math>c</math>. At <math>c</math>, <math>f</math> is differentiable iff <math>f_1(c) = f_2(c) = v</math> and <math>f_1'(c) = f_2'(c)</math>. \|\| [[differentiation rule for piecewise definition by interval]]
	\|}		\|}

	<section end="differentiation rules"/>		<section end="differentiation rules"/>

Vipul: /* Leibniz notation for derivative */

2021-09-24T14:17:09Z

Leibniz notation for derivative

← Older revision		Revision as of 14:17, 24 September 2021
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	<math>\! \frac{dy}{dx} \|_{x = x_0} := f'(x_0)</math>		<math>\! \frac{dy}{dx} \|_{x = x_0} := f'(x_0)</math>

	Note that the <math>dy/dx</math> notation does ''not'' mean that a number <math>dy</math> is being divided by a number <math>dx</math>. One way of justifying this notation is by expressing it as a limit of a [[difference quotient]]:		Note that the <math>dy/dx</math> notation does ''not'' mean that a number <math>dy</math> is being divided by a number <math>dx</math>. One way of justifying this notation is by expressing it as a limit of a [[difference quotient]]; here, <math>y_0 = f(x_0)</math>:

	<math>\! \frac{dy}{dx} \|_{x = x_0} := \lim_{x \to x_0} \frac{y - y_0}{x - x_0} = \lim_{x \to x_0} \frac{\Delta y}{\Delta x}</math>		<math>\! \frac{dy}{dx} \|_{x = x_0} := \lim_{x \to x_0} \frac{y - y_0}{x - x_0} = \lim_{x \to x_0} \frac{\Delta y}{\Delta x}</math>