Math 22 Implicit Differentiation

Implicit Differentiation

Consider the equation $x^{2}y=5$. To find ${\frac {dy}{dx}}$, we can rewrite the equation as $y={\frac {5}{x^{2}}}$, then differentiate as usual. ie: $y={\frac {5}{x^{2}}}=5x^{-2}$, so ${\frac {dy}{dx}}=-10x^{-3}$. This is called explicit differentiation.

However, sometimes, it is difficult to express $y$ as a function of $x$ explicitly. For example: $y^{2}-2x+4xy=5$

Therefore, we can use the procedure called implicit differentiation

Guidelines for Implicit Differentiation

 Consider an equation involving $x$ and $y$ in which $y$ is a differentiable function of $x$. 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}}
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 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}
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 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}}
.

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