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Implicit Differentiation Explained: Finding Slope & Equations of Tangent and Normal Lines

Learn how to find slope, equations of tangent, and normal lines using implicit differentiation technique with examples and insights. Explore higher-order derivatives for advanced applications.

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Implicit Differentiation Explained: Finding Slope & Equations of Tangent and Normal Lines

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  1. Implicit Differentiation Niagara Falls, NY & Canada Greg Kelly, Hanford High School, Richland, Washington Adapted by: Jon Bannon Siena College Photo by Vickie Kelly, 2003

  2. This is not a function, but it would still be nice to be able to find the slope. Do the same thing to both sides. 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. Find if . Substitute back into the equation. Higher Order Derivatives

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