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Implicit Differentiation: Tangent and Normal Lines

Learn how to find slope using chain rule and implicit differentiation. Solve for slopes of tangent and normal lines at a given point on a curve.

brittnee-noble
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Implicit Differentiation: Tangent and Normal Lines

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  1. Section 2.5 Implicit Differentiation

  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 x2 − xy + y2 = 7 at (−1, 2). Note product rule.

  5. The normal line at a point is perpendicular to the tangent line at the point.

  6. Find the equations of the lines tangent and normal to the curve x2 − xy + y2 = 7 at (−1, 2). tangent: normal:

  7. Find if . Higher Order Derivatives Substitute y′ back into the equation.

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