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Understanding Implicit Differentiation in AP Calculus AB/BC

This resource explores implicit differentiation, a crucial technique in calculus used for finding slopes of curves that cannot be solved for ( y ) explicitly. Using examples, the guide explains how to differentiate both sides of an equation with respect to ( x ), apply the chain rule, and solve for the derivative. It provides step-by-step instructions on utilizing the product rule and finding equations for tangent and normal lines to curves. This is essential for mastering calculus concepts, preparing for AP exams, and enhancing problem-solving skills.

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Understanding Implicit Differentiation in AP Calculus AB/BC

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  1. AP Calculus AB/BC 3.7 Implicit Differentiation, p. 157 Day 1

  2. Example 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 . See p. 159 Example This can’t be solved for y. Using the Product Rule This technique is called implicit differentiation.

  4. Example Distribute the sec2 and subtract 1.

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

  6. Find the equations of the lines tangent and normal to the curve at . tangent: normal: p Day 1

  7. AP Calculus AB/BC 3.7 Implicit Differentiation, p. 157 Day 2

  8. Find if . Substitute back into the equation. Higher Order Derivatives p Day 2

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