Difference between revisions of "Math 22 Chain Rule"
Jump to navigation
Jump to search
| Line 7: | Line 7: | ||
In another word, <math>\frac{d}{dx}[f(g(x))]=f'(g(x))\cdot g'(x)</math> | In another word, <math>\frac{d}{dx}[f(g(x))]=f'(g(x))\cdot g'(x)</math> | ||
| + | |||
| + | '''Example''': Find derivative of | ||
| + | |||
| + | '''1)''' <math>f(x)=\sqrt{x^2+3x-4}</math> | ||
| + | {| class = "mw-collapsible mw-collapsed" style = "text-align:left;" | ||
| + | !Solution: | ||
| + | |- | ||
| + | |<math style="vertical-align: -5px">f(x)=f(x)=\sqrt{x^2+3x-4}=(x^2+3x-4)^{\frac{1}{2}}</math> | ||
| + | |- | ||
| + | |<math>f'(x)=\frac{1}{2}\cdot (x^2+3x-4)^{\frac{1}{2} -1}\frac{d}{dx}[x^2+3x-4]</math> | ||
| + | |- | ||
| + | |<math=(x^2+3x-4)^{\frac{-1}{2}}(2x+3) | ||
| + | |} | ||
| + | |||
| + | '''2)''' <math>f(x)=(x^2+1)^100</math> | ||
| + | {| class = "mw-collapsible mw-collapsed" style = "text-align:left;" | ||
| + | !Solution: | ||
| + | |- | ||
| + | |<math style="vertical-align: -5px">f'(x)=100(x^2+1)^99 \frac{d}{dx}[x^2+1]</math> | ||
| + | |- | ||
| + | |<math>=100(x^2+1)^99 (2x)</math> | ||
| + | |} | ||
[[Math_22| '''Return to Topics Page''']] | [[Math_22| '''Return to Topics Page''']] | ||
'''This page were made by [[Contributors|Tri Phan]]''' | '''This page were made by [[Contributors|Tri Phan]]''' | ||
Revision as of 06:10, 23 July 2020
The Chain Rule
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 y=f(x)}
is a differentiable function 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 u}
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 u=g(x)}
is a
differentiable function 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 x}
, 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 y=f(g(x))}
is a differentiable function
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 x}
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 \frac{dy}{dx}=\frac{dy}{du}\cdot\frac{du}{dx}}
In another word, 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 \frac{d}{dx}[f(g(x))]=f'(g(x))\cdot g'(x)}
Example: Find 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)=\sqrt{x^2+3x-4}}
| Solution: |
|---|
| 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)=f(x)=\sqrt{x^2+3x-4}=(x^2+3x-4)^{\frac{1}{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)=\frac{1}{2}\cdot (x^2+3x-4)^{\frac{1}{2} -1}\frac{d}{dx}[x^2+3x-4]} |
| <math=(x^2+3x-4)^{\frac{-1}{2}}(2x+3) |
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^2+1)^100}
| Solution: |
|---|
| 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)=100(x^2+1)^99 \frac{d}{dx}[x^2+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 =100(x^2+1)^99 (2x)} |
This page were made by Tri Phan