Difference between revisions of "Math 22 Implicit Differentiation"
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==Implicit Differentiation== | ==Implicit Differentiation== | ||
| − | Consider the equation <math>x^2y=5</math>. To find <math>\frac{dy}{dx}</math>, we can rewrite the equation as <math>y=\frac{5}{x^2}</math>, then differentiate as usual. ie: <math>y=\frac{5}{x^2}=5x^{-2}</math>, so <math>\frac{dy}{dx}=-10x^{-3}</math> | + | Consider the equation <math>x^2y=5</math>. To find <math>\frac{dy}{dx}</math>, we can rewrite the equation as <math>y=\frac{5}{x^2}</math>, then differentiate as usual. ie: <math>y=\frac{5}{x^2}=5x^{-2}</math>, so <math>\frac{dy}{dx}=-10x^{-3}</math>. This is called explicit differentiation. However, sometimes, it is difficult to express <math>y</math> as a function of <math>x</math> explicitly. For example: <math>y^2-2x+4xy=5</math> |
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| + | Therefore, we can use the procedure called '''implicit differentiation''' | ||
Revision as of 09:12, 25 July 2020
Implicit Differentiation
Consider the equation 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=5} . 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}} , we can rewrite the equation 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 y=\frac{5}{x^2}} , then differentiate as usual. ie: 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=\frac{5}{x^2}=5x^{-2}} , so 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}=-10x^{-3}} . This is called explicit differentiation. However, sometimes, it is difficult to express 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} as a 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} explicitly. For example: 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^2-2x+4xy=5}
Therefore, we can use the procedure called implicit differentiation
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