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2014 Implicit Differentiation

2014 Implicit Differentiation. Calculus BC. Implicit Differentiation. Equation for a line: Explicit Form <One variable given explicitly in terms of the other> Implicit Form <Function implied by the equation>   Differentiate the Explicit < Explicit : , y is function of x >

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2014 Implicit Differentiation

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  1. 2014 Implicit Differentiation Calculus BC

  2. Implicit Differentiation Equation for a line: Explicit Form <One variable given explicitly in terms of the other> Implicit Form <Function implied by the equation>   Differentiate the Explicit < Explicit: , y is function of x > Differentiation taking place with respect to x. The derivative is explicit also.

  3. Implicit Differentiation Equation of circle: To work explicitly; must work two equations Implicit Differentiation is a Short Cut - A method to handle equations that are not easily written explicitly. ( Usually non-functions)

  4. Implicit Differentiation Find the derivative with respect to x < Assuming - y is a differentiable function of x > Chain Rule Pretend y is some function like so becomes (A) (B) (C) Note: Use the Leibniz form. Leads to Parametric and Related Rates.

  5. Implicit Differentiation (D) Product Rule  (E) Chain Rule

  6. Implicit Differentiation To find implicitly. EX: Diff Both Sides of equation with respect to x Solve for

  7. EX 1: (a) Find the derivative at the point ( 5, 3 ) , at ( -1,-3 ) (b) Find where the curve has a horizontal tangent.  (c) Find where the curve has vertical tangents.

  8. Ex 2: < Folium of Descartes >

  9. Why Implicit? Explicit Form: < Folium of Descartes >

  10. Ex 2 Graph: < Folium of Descartes > Parametric Form: Plot the Folium of Descartes on your graphing calculator and determine the portion of the folium generated when (a) t < -1 ; (b) -1 < t  0 ; (c) t > 0

  11. 2nd Derivatives EX: Our friendly circle. Find the 2nd Derivative. NOTICE:The second derivative is in terms of x , y , AND dy /dx. The final step will be to substitute back the value of dy / dx into the second derivative.

  12. EX: Find the 2nd Derivative. 2nd Derivatives

  13. EX: Find the Third Derivative. Higher Derivatives

  14. Last update • 10/19/10 • p. 162 11 – 29 odd

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