Second derivative - Revision history

Vipul: /* Significance */

2011-12-18T03:15:04Z

Significance

← Older revision		Revision as of 03:15, 18 December 2011
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	A [[point of inflection]] is a point of geometric significance on the graph (where the function changes its sense of concavity), and corresponds to a point in the domain of the function where the second derivative changes sign. A point of inflection must correspond to a point of (strict) local maximum or minimum for the [[first derivative]].		A [[point of inflection]] is a point of geometric significance on the graph (where the function changes its sense of concavity), and corresponds to a point in the domain of the function where the second derivative changes sign. A point of inflection must correspond to a point of (strict) local maximum or minimum for the [[first derivative]].

			{\| class="sortable" border="1"
			! Method for constructing new functions from old !! In symbols !! Derivative in terms of old functions and their first, second derivatives !! Proof
			\|-
			\| [[pointwise sum of functions\|pointwise sum]] \|\| <math>f + g</math> is the function <math>x \mapsto f(x) + g(x)</math><br><math>f_1 + f_2 + \dots + f_n</math> is the function <math>x \mapsto f_1(x) + f_2(x) + \dots + f_n(x)</math> \|\| Sum of the second derivatives of the functions being added (''the second derivative of the sum is the sum of the second derivatives'')<br><math>\! f'' + g''</math><br><math>\! f_1'' + f_2'' + \dots + f_n''</math> \|\| [[repeated differentiation is linear]]
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			\| [[pointwise difference of functions\|pointwise difference]] \|\| <math>f - g</math> is the function <math>x \mapsto f(x) - g(x)</math>\|\| Difference of the second derivatives, i.e., <math>f'' - g''</math> \|\| [[repeated differentiation is linear]]
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			\| [[scalar multiple of function\|scalar multiple]] by a constant \|\| <math>af</math> is the function <math>x \mapsto af(x)</math> where <math>a</math> is a real number \|\| <math>x \mapsto af''(x)</math> \|\| [[repeated differentiation is linear]]
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			\| [[pointwise product of functions\|pointwise product]] \|\| <math>f \cdot g</math> (sometimes denoted <math>fg</math>) is the function <math>x \mapsto f(x)g(x)</math><br><math>f_1 \cdot f_2 \cdot \dots f_n</math> (sometimes denoted <math>f_1f_2\dots f_n</math> is the function <math>x \mapsto f_1(x)f_2(x) \dots f_n(x)</math> \|\| For two functions, <math>x \mapsto f''(x)g(x) + 2f'(x)g'(x) + f(x)g''(x)</math><br>For multiple functions, more complicated \|\| [[product rule for higher derivatives]]
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			\| [[pointwise quotient of functions\|pointwise quotient]] \|\| <math>f/g</math> is the function <math>x \mapsto f(x)/g(x)</math> \|\| {{fillin}} \|\| [[quotient rule for higher derivatives]]
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			\| [[composite of two functions]] \|\| <math>f \circ g</math> is the function <math>x \mapsto f(g(x))</math> \|\| <math>x \mapsto f''(g(x))(g'(x))^2 + f'(g(x))g''(x)</math> \|\| [[chain rule for higher derivatives]]
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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{-f''(f^{-1}(x))}{(f'(f^{-1}(x)))^3}</math> \|\| [[higher derivatives of inverse function]]
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			\| [[piecewise definition of functions\|piecewise definition]] \|\| {{fillin}} \|\| {{fillin}} \|\| [[differentiation rule for piecewise definition by interval]]
			\|}

Vipul at 03:09, 18 December 2011

2011-12-18T03:09:27Z

← Older revision		Revision as of 03:09, 18 December 2011
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	<math>\! \frac{d^2y}{dx^2} \|_{x = x_0} := f''(x_0)</math>		<math>\! \frac{d^2y}{dx^2} \|_{x = x_0} := f''(x_0)</math>

			==Significance==

			===Significance of sign on intervals===

			{\| class="sortable" border="1"
			! Loose statement !! Precise statements
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			\| [[concave up function]] means derivative is increasing means second derivative is positive \|\| [[positive second derivative implies concave up]]<br>{{fillin}}
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			\| [[concave down function]] means derivative is decreasing means second derivative is negative \|\| [[negative second derivative implies concave down]]
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			\| [[linear function]] (or [[constant function]]) means derivative is constant means second derivative is zero \|\|
			\|}

			===Significance of sign at points===

			A [[point of inflection]] is a point of geometric significance on the graph (where the function changes its sense of concavity), and corresponds to a point in the domain of the function where the second derivative changes sign. A point of inflection must correspond to a point of (strict) local maximum or minimum for the [[first derivative]].

Vipul at 03:04, 18 December 2011

2011-12-18T03:04:50Z

← Older revision		Revision as of 03:04, 18 December 2011
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	<math>\! f'' := (f')'</math>		<math>\! f'' := (f')'</math>

			==Leibniz notation for second derivative==

			Suppose <math>f</math> is a function, and <math>x,y</math> are variables related by <math>y := f(x)</math>. Here, <math>x</math> is an ''independent variable'' and <math>y</math> is the ''dependent variable'' (with the dependency being described by the function <math>f</math>). We then define:

			<math>\frac{d^2y}{dx^2} := f''(x)</math>

			This can also be written as:

			<math>\frac{d^2}{dx^2}[f(x)]</math>

			In particular, <math>d^2y/dx^2</math> is a ''function'' of <math>x</math>. Its value at <math>x = x_0</math> is defined as <math>f''(x_0)</math> and is denoted as follows:

			<math>\! \frac{d^2y}{dx^2} \|_{x = x_0} := f''(x_0)</math>

Vipul: /* Definition as a limit expression */

2011-12-18T03:02:03Z

Definition as a limit expression

← Older revision		Revision as of 03:02, 18 December 2011
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	More explicitly, this can be written as:		More explicitly, this can be written as:

	<math>\! \lim_{x \to x_0} \frac{1}{x - x_0}\left[\lim_{x_1 \to x} \frac{f(x_1) - f(x)}{x_1 - x} - \lim_{x_2 \to x} \frac{f(x_2) - f(x_0)}{x_2 - x_0}\right]</math>		<math>\! \lim_{x \to x_0} \frac{1}{x - x_0}\left[\lim_{x_1 \to x} \frac{f(x_1) - f(x)}{x_1 - x} - \lim_{x_2 \to x_0} \frac{f(x_2) - f(x_0)}{x_2 - x_0}\right]</math>

	==Definition as a function==		==Definition as a function==

Vipul at 03:01, 18 December 2011

2011-12-18T03:01:39Z

← Older revision		Revision as of 03:01, 18 December 2011
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	==Definition at a point==		==Definition at a point==

	The '''second derivative''' of a function <math>f</math> at a point <math>x_0</math>, denoted <math>f''(x_0)</math>, is defined as the [[derivative]] at the point <math>x_0</math> of the function defined as the [[derivative]] <math>f'</math>		===Definition in terms of first derivative===

			The '''second derivative''' of a function <math>f</math> at a point <math>x_0</math>, denoted <math>\! f''(x_0)</math>, is defined as the [[derivative]] at the point <math>x_0</math> of the function defined as the [[derivative]] <math>f'</math>

	Note that the ''first'' differentiation operation must be performed, not just at the point, but at all points near it, so that we have a ''function'' for the first derivative around the point, which we can then differentiate to calculate the second derivative ''at'' the point. It is ''not'' good enough to calculate the first derivative only ''at'' the particular point (i.e., to calculate ''only'' <math>f'(x_0)</math>) and then proceed to differentiate that; we need the value of the first derivative at nearby points too.		Note that the ''first'' differentiation operation must be performed, not just at the point, but at all points near it, so that we have a ''function'' for the first derivative around the point, which we can then differentiate to calculate the second derivative ''at'' the point. It is ''not'' good enough to calculate the first derivative only ''at'' the particular point (i.e., to calculate ''only'' <math>f'(x_0)</math>) and then proceed to differentiate that; we need the value of the first derivative at nearby points too.

			===Definition as a limit expression===

			The '''second derivative''' of a function <math>f</math> at a point <math>x_0</math>, denoted <math>\! f''(x_0)</math>, is defined as follows:

			<math>\! \lim_{x \to x_0} \frac{f'(x) - f'(x_0)}{x - x_0}</math>

			More explicitly, this can be written as:

			<math>\! \lim_{x \to x_0} \frac{1}{x - x_0}\left[\lim_{x_1 \to x} \frac{f(x_1) - f(x)}{x_1 - x} - \lim_{x_2 \to x} \frac{f(x_2) - f(x_0)}{x_2 - x_0}\right]</math>

	==Definition as a function==		==Definition as a function==

Vipul: Created page with "==Definition at a point== The '''second derivative''' of a function $f$ at a point $x_0$ , denoted $f''(x_0)$ , is defined as the derivative a..."

2011-08-29T13:16:05Z

Created page with "==Definition at a point== The '''second derivative''' of a function <math>f</math> at a point <math>x_0</math>, denoted <math>f''(x_0)</math>, is defined as the derivative a..."

New page

==Definition at a point==

The '''second derivative''' of a function <math>f</math> at a point <math>x_0</math>, denoted <math>f''(x_0)</math>, is defined as the [[derivative]] at the point <math>x_0</math> of the function defined as the [[derivative]] <math>f'</math>

Note that the ''first'' differentiation operation must be performed, not just at the point, but at all points near it, so that we have a ''function'' for the first derivative around the point, which we can then differentiate to calculate the second derivative ''at'' the point. It is ''not'' good enough to calculate the first derivative only ''at'' the particular point (i.e., to calculate ''only'' <math>f'(x_0)</math>) and then proceed to differentiate that; we need the value of the first derivative at nearby points too.

==Definition as a function==

The '''second derivative''' of a function at a point is defined as the [[derivative]] of the [[derivative]] of the function. For a function <math>f</math>, the second derivative <math>f''</math> is defined as:

<math>\! f'' := (f')'</math>