Implicit Differentiation

JSFH Obtain the derivative of the function LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSJ5RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEwJkFwcGx5RnVuY3Rpb247RicvRjhRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZCLyUpc3RyZXRjaHlHRkIvJSpzeW1tZXRyaWNHRkIvJShsYXJnZW9wR0ZCLyUubW92YWJsZWxpbWl0c0dGQi8lJ2FjY2VudEdGQi8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRlEtSShtZmVuY2VkR0YkNiQtRiM2JC1GLDYlUSJ4RidGNEY3Rj5GPkY+RitGPg== defined implicitly by the equation 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

LywoKiYiIiQiIiIpSSJ4RzYiIiIjRiZGJiooIiImRiZGKEYmSSJ5R0YpRiYhIiIqJiIiKEYmKUYtRipGJkYmIiM2

plots[implicitplot](, x = -5 .. 5, y = -5 .. 5, color = "Red");

JSFH JSFH

JSFH Control-drag the equation. Context Menu: Plots\342\211\2732-D Implicit Plot\342\211\273LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIixGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMzMzMzMzM2VtRictRiw2JVEieUYnRi9GMkY5 JSFH JSFH Explore the branches Control-drag the equation Context Menu: Solve\342\211\273Obtain Solutions for\342\211\273LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEieUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= Context Menu: Plots\342\211\2732-D Plot JSFH JSFH

JSFH Control-drag equation. Context Menu: Differentiate Implicitly JSFH JSFH JSFH

JSFH Tools\342\211\273Tasks\342\211\273Browse: Calculus - Differential\342\211\273Derivatives\342\211\273Implicit Differentiation\342\211\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 JSFH Implicit Differentiation Enter an equation in two variables: use DocumentTools in # Enter Maple commands to be executed when the specified # action is carried out on the component. # Use: # GetProperty( component_name, attribute_name ) # and # SetProperty( component_name, attribute_name, value ) # to access any of the attributes of the component. # See ?CustomizingComponents for more information. module() SetProperty('MathContainer8', 'value', NULL); SetProperty('MathContainer9', 'value', NULL); SetProperty('MathContainer10', 'value', NULL); SetProperty('MathContainer11', 'value', NULL); SetProperty('MathContainer12', 'value', NULL); SetProperty('MathContainer13', 'value', NULL); SetProperty('MathContainer14', 'value', NULL); SetProperty('MathContainer15', 'value', NULL); end module; end use; Dependent variable: Independent variable: Implicit Derivative: use DocumentTools in # Enter Maple commands to be executed when the specified # action is carried out on the component. # Use: # GetProperty( component_name, attribute_name ) # and # SetProperty( component_name, attribute_name, value ) # to access any of the attributes of the component. # See ?CustomizingComponents for more information. module() local Var_ID_1, Var_ID_2, Var_ID_3, Var_ID_4; _EnvDisableGlobalBindings:=true: Do(Var_ID_1 = %MathContainer8); Do(Var_ID_2 = %MathContainer9); Do(Var_ID_3 = %MathContainer10); Var_ID_4 := implicitdiff(Var_ID_1, Var_ID_2, Var_ID_3); SetProperty('MathContainer11','value', cat(d,Var_ID_2)/cat(d,Var_ID_3) = Var_ID_4); end module; end use; Stepwise Calculation Make dependent variable explicit: use DocumentTools in # Enter Maple commands to be executed when the specified # action is carried out on the component. # Use: # GetProperty( component_name, attribute_name ) # and # SetProperty( component_name, attribute_name, value ) # to access any of the attributes of the component. # See ?CustomizingComponents for more information. module() local Var_ID_1, Var_ID_2, Var_ID_3, Var_ID_4, Var_ID_5; _EnvDisableGlobalBindings:=true: Do(Var_ID_1 = %MathContainer8); Do(Var_ID_2 = %MathContainer9); Do(Var_ID_3 = %MathContainer10); Var_ID_4 := implicitdiff(Var_ID_1, Var_ID_2, Var_ID_3); Var_ID_5 := eval(Var_ID_1, Var_ID_2 = Var_ID_2(Var_ID_3) ); SetProperty('MathContainer12', 'value', Var_ID_5); end module; end use; Differentiate with respect to independent variable: use DocumentTools in # Enter Maple commands to be executed when the specified # action is carried out on the component. # Use: # GetProperty( component_name, attribute_name ) # and # SetProperty( component_name, attribute_name, value ) # to access any of the attributes of the component. # See ?CustomizingComponents for more information. module() local Var_ID_1, Var_ID_2, Var_ID_3, Var_ID_4, Var_ID_5, Var_ID_6; _EnvDisableGlobalBindings:=true: Do(Var_ID_1 = %MathContainer8); Do(Var_ID_2 = %MathContainer9); Do(Var_ID_3 = %MathContainer10); Var_ID_4 := implicitdiff(Var_ID_1, Var_ID_2, Var_ID_3); Var_ID_5 := eval(Var_ID_1, Var_ID_2 = Var_ID_2(Var_ID_3) ); Var_ID_6 := diff(Var_ID_5, Var_ID_3); SetProperty('MathContainer13', 'value', Var_ID_6); end module; end use; use DocumentTools in # Enter Maple commands to be executed when the specified # action is carried out on the component. # Use: # GetProperty( component_name, attribute_name ) # and # SetProperty( component_name, attribute_name, value ) # to access any of the attributes of the component. # See ?CustomizingComponents for more information. module() local Var_ID_1, Var_ID_2, Var_ID_3, Var_ID_5; Do(Var_ID_1 = %MathContainer8); if type(Var_ID_1,equation) then Var_ID_1 := lhs(Var_ID_1) - rhs(Var_ID_1); end if; Do(Var_ID_2 = %MathContainer9); Do(Var_ID_3 = %MathContainer10); Var_ID_5 := eval(Var_ID_1, Var_ID_2 = Var_ID_2(Var_ID_3) ); Student:-Calculus1:-DiffTutor(Var_ID_5, Var_ID_3); end module; end use; Isolate Derivative: use DocumentTools in # Enter Maple commands to be executed when the specified # action is carried out on the component. # Use: # GetProperty( component_name, attribute_name ) # and # SetProperty( component_name, attribute_name, value ) # to access any of the attributes of the component. # See ?CustomizingComponents for more information. module() local Var_ID_1, Var_ID_2, Var_ID_3, Var_ID_4, Var_ID_5, Var_ID_6, Var_ID_7; _EnvDisableGlobalBindings:=true: Do(Var_ID_1 = %MathContainer8); Do(Var_ID_2 = %MathContainer9); Do(Var_ID_3 = %MathContainer10); Var_ID_4 := implicitdiff(Var_ID_1, Var_ID_2, Var_ID_3); Var_ID_5 := eval(Var_ID_1, Var_ID_2 = Var_ID_2(Var_ID_3) ); Var_ID_6 := diff(Var_ID_5, Var_ID_3); Var_ID_7 := isolate(Var_ID_6, diff(Var_ID_2(Var_ID_3), Var_ID_3)); SetProperty('MathContainer14', 'value', Var_ID_7); end module; end use; Make independent variable implicit: use DocumentTools in # Enter Maple commands to be executed when the specified # action is carried out on the component. # Use: # GetProperty( component_name, attribute_name ) # and # SetProperty( component_name, attribute_name, value ) # to access any of the attributes of the component. # See ?CustomizingComponents for more information. module() local Var_ID_1, Var_ID_2, Var_ID_3, Var_ID_4, Var_ID_5, Var_ID_6, Var_ID_7, Var_ID_8; _EnvDisableGlobalBindings:=true: Do(Var_ID_1 = %MathContainer8); Do(Var_ID_2 = %MathContainer9); Do(Var_ID_3 = %MathContainer10); Var_ID_4 := implicitdiff(Var_ID_1, Var_ID_2, Var_ID_3); Var_ID_5 := eval(Var_ID_1, Var_ID_2 = Var_ID_2(Var_ID_3) ); Var_ID_6 := diff(Var_ID_5, Var_ID_3); Var_ID_7 := isolate(Var_ID_6, diff(Var_ID_2(Var_ID_3), Var_ID_3)); Var_ID_8 := simplify(eval(rhs(Var_ID_7), Var_ID_2(Var_ID_3)=Var_ID_2)); SetProperty('MathContainer15', 'value', cat(d,Var_ID_2)/cat(d,Var_ID_3) = Var_ID_8); end module; end use; JSFH

JSFH Control-drag the equation. Context Menu: Evaluate at a Point \342\211\273 Change LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEieUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= to LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEieUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUYsNiVRInhGJ0YvRjIvRjNRJ25vcm1hbEYnRj1GPQ== Context Menu: Differentiate\342\211\273LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= Context Menu: Solve\342\211\273Isolate Expression for \342\211\273 diff(y(x),x) JSFH JSFH JSFH JSFH JSFH JSFH JSFH JSFH

JSFH JSFH JSFH JSFH JSFH JSFH JSFH JSFH JSFH JSFH JSFH JSFH JSFH JSFH JSFH JSFH JSFH JSFH JSFH JSFH JSFH JSFH JSFH JSFH JSFH JSFH LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= JSFH LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= JSFH