Difference between revisions of "Math 22 Implicit Differentiation"
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| − | + | '''Example''': Find <math>\frac{dy}{dx}</math> of | |
| + | |||
| + | '''1)''' <math>y^3=5x^3+8x</math> | ||
| + | {| class = "mw-collapsible mw-collapsed" style = "text-align:left;" | ||
| + | !Solution: | ||
| + | |- | ||
| + | |<math>\frac{d}{dx}[y^3]=\frac{d}{dx}[5x^3+8x]</math> | ||
| + | |- | ||
| + | |<math>=3y^2\frac{dy}{dx}=\frac{d}{dx}[5x^3]+\frac{d}{dx}[8x]</math> | ||
| + | |- | ||
| + | |<math>=3y^2\frac{dy}{dx}=15x^2+8</math> | ||
| + | |- | ||
| + | |<math>=\frac{dy}{dx}=\frac{15x^2+8}{3y^2}</math> | ||
| + | |} | ||
| + | |||
| + | '''2)''' <math>x^2y+2xy+7=6x</math> | ||
| + | {| class = "mw-collapsible mw-collapsed" style = "text-align:left;" | ||
| + | !Solution: | ||
| + | |- | ||
| + | |<math>\frac{d}{dx}[x^2y+2xy+7]=\frac{d}{dx}[6x]</math> | ||
| + | |- | ||
| + | |<math>=\frac{d}{dx}[x^2y]+\frac{d}{dx}[2xy]+\frac{d}{dx}[7]=6</math> | ||
| + | |- | ||
| + | |<math>=[2xy+x^2\frac{dy}{dx}]+[2y+2x\frac{dy}{dx}]+0=6</math> | ||
| + | |- | ||
| + | |<math>=x^2\frac{dy}{dx}+2x\frac{dy}{dx}=6-2xy-2y</math> | ||
| + | |- | ||
| + | |<math>=(x^2+2x)\frac{dy}{dx}=6-2xy-2y</math> | ||
| + | |- | ||
| + | |<math>=\frac{dy}{dx}=\frac{6-2xy-2y}{x^2+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]]''' | ||
Latest revision as of 07:28, 27 July 2020
Implicit Differentiation
Consider the equation . To find , we can rewrite the equation as , then differentiate as usual. ie: , so . This is called explicit differentiation.
However, sometimes, it is difficult to express as a function of explicitly. For example:
Therefore, we can use the procedure called implicit differentiation
Guidelines for Implicit Differentiation
Consider an equation involving and in which is a differentiable function of . You can use the steps below to find 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}} . 1. Differentiate both sides of the equation with respect to 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} . 2. Collect all terms involving 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}} on the left side of the equation and move all other terms to the right side of the equation 3. Factor 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}} out of the left side of the equation. 4. Solve 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 \frac{dy}{dx}} by dividing both sides of the equation by the left-hand factor that does not contain 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}} .
Example: Find 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}}
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 y^3=5x^3+8x}
| 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 \frac{d}{dx}[y^3]=\frac{d}{dx}[5x^3+8x]} |
| 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 =3y^2\frac{dy}{dx}=\frac{d}{dx}[5x^3]+\frac{d}{dx}[8x]} |
| 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 =3y^2\frac{dy}{dx}=15x^2+8} |
| 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{15x^2+8}{3y^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 x^2y+2xy+7=6x}
| 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 \frac{d}{dx}[x^2y+2xy+7]=\frac{d}{dx}[6x]} |
| 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}[x^2y]+\frac{d}{dx}[2xy]+\frac{d}{dx}[7]=6} |
| 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 =[2xy+x^2\frac{dy}{dx}]+[2y+2x\frac{dy}{dx}]+0=6} |
| 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^2\frac{dy}{dx}+2x\frac{dy}{dx}=6-2xy-2y} |
| 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^2+2x)\frac{dy}{dx}=6-2xy-2y} |
| 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{6-2xy-2y}{x^2+2x}} |
This page were made by Tri Phan