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Implicit Differentiation

Implicit Differentiation. Section 3.7a. Consider the equation:. Is this a function?. Is the equation differentiable?. If so, how do we differentiate?. We use implicit differentiation , so named b/c the functions are defined implicitly (hidden) within the original equation .

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Implicit Differentiation

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  1. Implicit Differentiation Section 3.7a

  2. Consider the equation: Is this a function? Is the equation differentiable? If so, how do we differentiate?

  3. We use implicit differentiation, so named b/c the functions are defined implicitly (hidden) within the original equation

  4. Implicit Differentiation Process  We treat y as a differentiable function of x!!! • Differentiate both sides of the equation with • respect to x. Because of the Chain Rule, any time differentiating any term containing a y, also multiply by dy/dx!!! 2. Collect the terms with dy/dx on one side of the equation. 3. Factor out dy/dx. 4. Solve for dy/dx.

  5. Initial Guided Practice Find dy/dx: Implicit Differentiation: Solve for the derivative: Does this answer make sense graphically?

  6. Initial Guided Practice Find the slope of the circle at First, find the slope of any point on the circle via implicit differentiation: Slope at the given point: Again, verify this answer graphically!

  7. Initial Guided Practice Show that the slope dy/dx is defined at every point on the graph of Imp Diff: This formula for dy/dx is defined at every point (x,y) except for those points at which cos(y)=2… Which never happens!!!

  8. We can use all of this new info to expand the POWER of the POWER RULE!!!  A proof: First, let p and q be integers with q > 0 and suppose that: Subst. for y Law of Exp. Imp. Diff.! Final Answer!

  9. The EXPANDED Power Rule: If n is any rational number, then (If n < 1, then the derivative does not exist at x = 0) WHY NOT??? Note: Before, the power rule worked for integers only…  now, it works for any rational number power (i.e., fraction) What is the derivative of the square root function???

  10. Additional Guided Practice For each of the following, find dy/dx.

  11. Additional Guided Practice For each of the following, find dy/dx.

  12. Additional Guided Practice For each of the following, find dy/dx.

  13. Additional Guided Practice For each of the following, find dy/dx.

  14. Additional Guided Practice For each of the following, find dy/dx.

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