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Implicit Differentiation

Implicit Differentiation. This is not a function, but it would still be nice to be able to find the slope. Note use of chain rule. 1 Differentiate both sides w.r.t. x . 2 Solve for . This can’t be solved for y . This technique is called implicit differentiation.

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Implicit Differentiation

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  1. Implicit Differentiation

  2. This is not a function, but it would still be nice to be able to find the slope. Note use of chain rule.

  3. 1 Differentiate both sides w.r.t. x. 2 Solve for . This can’t be solved for y. This technique is called implicit differentiation.

  4. We need the slope. Since we can’t solve for y, we use implicit differentiation to solve for . Find the equations of the lines tangent and normal to the curve at . Note product rule.

  5. Find the equations of the lines tangent and normal to the curve at . tangent: normal:

  6. Examples: • x2y3 – xy = 10 • x2+3xy+y2 = -1 • x2+y2 = 25 • y3-x2 = x + y • 3y + ln y = 4ex • x2y= y3x + 5y + x • 25x2 + 8x – 16y2 - 4y – 9 = 0

  7. More examples: Find dy/dx • y = sin x + cos y • x2y + y3 + 3 = xy • xy3 + xy = 6 • ex + cos y = ln y6 • x2+ 2xy + y4 = 1 • x2y2 – ex + 2y = 4 • sin x/y = ½ • cos (x+y) + sin (x+y) = 1/3 • ecos x + esin y = 1/4

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