Difference between revisions of "Math 22 Higher-Order Derivative"

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(Created page with "==Higher-Order Derivatives== The "standard" derivative <math>f'(x)</math> is called the first derivative of <math>f(x)</math>. The derivative of <math>f'(x)</math> is the se...")
 
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Note: The 3rd derivative of <math>f(x)</math> is <math>f'''(x)</math>. However, we simply denote the <math>n^{th}</math> derivative as <math>f^{(n)}(x)</math> for <math>n\ge 4</math>
 
Note: The 3rd derivative of <math>f(x)</math> is <math>f'''(x)</math>. However, we simply denote the <math>n^{th}</math> derivative as <math>f^{(n)}(x)</math> for <math>n\ge 4</math>
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'''Example''': Find the first four derivative of
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'''1)''' <math>f(x)=x^4+5x^3-2x^2+6</math>
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{| class = "mw-collapsible mw-collapsed" style = "text-align:left;"
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!Solution: &nbsp;
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|-
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|f'(x)=4x^3+15x^2-4x
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|-
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|f''(x)=12x^2+30x-4
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|-
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|f'''(x)=24x+30
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|-
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|f^{(4)}(x)=24
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|}
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'''2)''' <math>f(x)=(x^3+1)(x^2+3)</math>
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{| class = "mw-collapsible mw-collapsed" style = "text-align:left;"
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!Solution: &nbsp;
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|-
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|It is better to rewrite <math>f(x)=(x^3+1)(x^2+3)=x^5+3x^3+x^2+3</math>
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|-
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|Then, <math>f'(x)=5x^4+9x^3+2x</math>
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|-
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|<math>f''(x)=20x^3+27x^2+2</math>
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|-
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|<math>f'''(x)=60x^2+54x</math>
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|-
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|<math>f^{(4)}(x)=120x+54</math>
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|}
  
 
==Notes==
 
==Notes==

Revision as of 08:35, 25 July 2020

Higher-Order Derivatives

 The "standard" derivative  is called the first derivative of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x)}
. The derivative of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f'(x)}
 is the second derivative ofFailed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x)}
, denoted by Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f''(x).}

 By continuing this process, we obtain higher-order derivative of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x)}
.

Note: The 3rd derivative of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x)} is Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f'''(x)} . However, we simply denote the Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n^{th}} derivative as Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f^{(n)}(x)} for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n\ge 4}

Example: Find the first four derivative of

1) Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x)=x^4+5x^3-2x^2+6}

Solution:  
f'(x)=4x^3+15x^2-4x
f(x)=12x^2+30x-4
f(x)=24x+30
f^{(4)}(x)=24

2) Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x)=(x^3+1)(x^2+3)}

Solution:  
It is better to rewrite Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x)=(x^3+1)(x^2+3)=x^5+3x^3+x^2+3}
Then, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f'(x)=5x^4+9x^3+2x}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f''(x)=20x^3+27x^2+2}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f'''(x)=60x^2+54x}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f^{(4)}(x)=120x+54}

Notes

If Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x)} is the position function, then Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f'(x)} is the velocity function and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f''(x)} is the acceleration function.

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