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> | ||
+ | |||
+ | '''Example''': Find the first four derivative of | ||
+ | |||
+ | '''1)''' <math>f(x)=x^4+5x^3-2x^2+6</math> | ||
+ | {| class = "mw-collapsible mw-collapsed" style = "text-align:left;" | ||
+ | !Solution: | ||
+ | |- | ||
+ | |f'(x)=4x^3+15x^2-4x | ||
+ | |- | ||
+ | |f''(x)=12x^2+30x-4 | ||
+ | |- | ||
+ | |f'''(x)=24x+30 | ||
+ | |- | ||
+ | |f^{(4)}(x)=24 | ||
+ | |} | ||
+ | |||
+ | '''2)''' <math>f(x)=(x^3+1)(x^2+3)</math> | ||
+ | {| class = "mw-collapsible mw-collapsed" style = "text-align:left;" | ||
+ | !Solution: | ||
+ | |- | ||
+ | |It is better to rewrite <math>f(x)=(x^3+1)(x^2+3)=x^5+3x^3+x^2+3</math> | ||
+ | |- | ||
+ | |Then, <math>f'(x)=5x^4+9x^3+2x</math> | ||
+ | |- | ||
+ | |<math>f''(x)=20x^3+27x^2+2</math> | ||
+ | |- | ||
+ | |<math>f'''(x)=60x^2+54x</math> | ||
+ | |- | ||
+ | |<math>f^{(4)}(x)=120x+54</math> | ||
+ | |} | ||
==Notes== | ==Notes== |
Revision as of 08:35, 25 July 2020
Higher-Order Derivatives
The "standard" derivative is called the first derivative of . The derivative of is the second derivative of, denoted by By continuing this process, we obtain higher-order derivative of .
Note: The 3rd derivative of is . However, we simply denote the derivative as for
Example: Find the first four derivative of
1)
Solution: |
---|
f'(x)=4x^3+15x^2-4x |
f(x)=12x^2+30x-4 |
f(x)=24x+30 |
f^{(4)}(x)=24 |
2)
Solution: |
---|
It is better to rewrite |
Then, |
Notes
If is the position function, then is the velocity function and is the acceleration function.
This page were made by Tri Phan