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Ms. Battaglia AB/BC Calculus

2.5 Implicit Differentiation Objective: Distinguish between functions written in implicit form and explicit form. Use implicit differentiation to find the derivative of a function. Ms. Battaglia AB/BC Calculus. Implicit & Explicit Functions.

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Ms. Battaglia AB/BC Calculus

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  1. 2.5 Implicit DifferentiationObjective: Distinguish between functions written in implicit form and explicit form. Use implicit differentiation to find the derivative of a function. Ms. Battaglia AB/BC Calculus

  2. Implicit & Explicit Functions • Up to this point, most functions have been expressed in explicit form. Ex: y=3x2– 5 • The variable y is explicitly written as a function of x. • How would you find dy/dx for x2 – 2y3 + 4y = 2? You can use implicit differentiation (apply the Chain Rule, because you are assuming that y is defined implicitly as a differentiable function of x).

  3. Differentiating with Respect to x Variables agree  Variables disagree 

  4. Differentiating with Respect to x

  5. Guidelines for Implicit Differentiation • Differentiate both sides of the equation with respect to x. • Collect all terms involving dy/dx on the left side of the equation and move all other terms to the right side of the equation. • Factor dy/dx out of the left side of the equation. • Solve for dy/dx.

  6. Implicit Differentiation Find dy/dx given that y3 + y2 – 5y – x2 = -4

  7. Representing a Graph by Differentiable Functions If possible, represent y as a differentiable function of x. a. x2 + y2 = 0 b. x2 + y2 = 1 c. x + y2 = 1

  8. Finding the Slope of a Graph Implicitly Determine the slope of the tangent line to the graph of x2 + 4y2 = 4 at the point

  9. Finding the Slope of a Graph Implicitly Determine the slope of the graph of 3(x2 + y2)2 = 100xy at the point (3,1).

  10. Determining a Differentiable Function Find dy/dx implicitly for the equation siny=x

  11. Finding the 2nd Derivative Given x2 + y2 = 25, find . Evaluate the 1st and 2nd derivatives at the point (-3,4).

  12. Finding a Tangent Line to a Graph Find the tangent line to the graph given by x2 (x2 + y2) = y2 at the point .

  13. Classwork/Homework • Read 2.5, Page 146 #7, 11, 21, 27, 30, 45, 47, 51

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