Derivative ImplicitlyDerivative Implicitly

Date:12/1/07Date:12/1/07

Long ZhaoLong Zhao

Teacher:Ms.DelacruzTeacher:Ms.Delacruz

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

Find of Find of

x²+5x+3y²=21x²+5x+3y²=21Step 1:Take derivative on both sidesStep 1:Take derivative on both sides (x²)+ (3y²)= (21)(x²)+ (3y²)= (21)

Step 2:Get the derivativeStep 2:Get the derivative2x+6y =02x+6y =0

Step 3:Solve forStep 3:Solve for =-2x/6y=-2x/6y

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

We find the derivativeWe find the derivative

The equation of the tangent line is: The equation of the tangent line is:

y - 1 = 0.25 (x - 2) y - 1 = 0.25 (x - 2)

Source:http://archives.math.utk.edu/Source:http://archives.math.utk.edu/visual.calculus/3/implicit.5/index.htmlvisual.calculus/3/implicit.5/index.html

An example of an implicit function, for which An example of an implicit function, for which implicit differentiation might be easier than implicit differentiation might be easier than attempting to use explicit differentiation, isattempting to use explicit differentiation, isx³+2y²=8x³+2y²=8In order to differentiate this explicitly, one would In order to differentiate this explicitly, one would have to obtain (via algebra)have to obtain (via algebra)Y=±Y=±

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and then differentiate this function. This creates and then differentiate this function. This creates two derivatives: one for two derivatives: one for yy > 0 and another for > 0 and another for yy < 0.< 0.

One might find it substantially easier to implicitly One might find it substantially easier to implicitly differentiate the implicit function;differentiate the implicit function;

One might find it substantially easier to One might find it substantially easier to implicitly differentiate the implicit implicitly differentiate the implicit function;4x³+4y =0function;4x³+4y =0

thus, thus,

Source:wikipediaSource:wikipedia

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