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Derivative Implicitly Date:12/1/07 Long Zhao Teacher:Ms.Delacruz

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In calculus, a method called implicit differentiation can be applied to implicitly defined functions. This method is an application of the chain rule allowing one to calculate the derivative of a function given implicitly. calculuschain rulecalculuschain rule As explained in the introduction, y can be given as a function of x implicitly rather than explicitly. When we have an equation R(x,y) = 0, we may be able to solve it for y and then differentiate. However, sometimes it is simpler to differentiate R(x,y) with respect to x and then solve for dy / dx.Source:Wikipedia

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Find of x²+5x+3y²=21 Step 1:Take derivative on both sides (x²)+ (3y²)= (21) Step 2:Get the derivative 2x+6y =0 Step 3:Solve for =-2x/6y

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To find the equation of the tangent line of To find the equation of the tangent line of

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We find the derivative

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The equation of the tangent line is: y - 1 = 0.25 (x - 2) Source:http://archives.math.utk.edu/v isual.calculus/3/implicit.5/index.html

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An example of an implicit function, for which implicit differentiation might be easier than attempting to use explicit differentiation, is x³+2y²=8 In order to differentiate this explicitly, one would have to obtain (via algebra) Y=±

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and then differentiate this function. This creates two derivatives: one for y > 0 and another for y 0 and another for y < 0. One might find it substantially easier to implicitly differentiate the implicit function; One might find it substantially easier to implicitly differentiate the implicit function;4x³+4y =0 thus,Source:wikipedia

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Example: Sec 3.7: Implicit Differentiation. Example: In some cases it is possible to solve such an equation for as an explicit function In many cases.

Example: Sec 3.7: Implicit Differentiation. Example: In some cases it is possible to solve such an equation for as an explicit function In many cases.

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