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Lesson 11: Implicit Differentiation Applications

1. Review of Implicit Differentiation

When y is not solved explicitly, differentiate both sides w.r.t. \(x\), treating \(y\) as \(y(x)\):

Example: \(x^2 + y^2 = 25\) (circle)

\(2x + 2y dy/dx = 0 \rightarrow dy/dx = -x/y\)

At point \((3,4)\): slope = -3/4

Exercise 1

For \(x^2 + xy + y^2 = 19\), find \(dy/dx\) at \((3,2)\):
\(2x + y + x dy/dx + 2y dy/dx = 0\)
\(dy/dx (x + 2y) =\)
\(dy/dx =\) /

2. Implicit Applications: Tangent Lines

Find slope → write equation of tangent line.

Example: \(x^3 + y^2 = 9xy\) at \((2,1)\)

\(3x^2 + 3y^2 dy/dx = 9y + 9x dy/dx\)

\(dy/dx = (9y - 3x^2)/(3y^2 - 9x)\)

At \((2,1)\): slope = \((9-12)/(3-18) = (-3)/(-15) = 1/5\)

Exercise 2

For \(x^2 + y^2 = 100\), what is \(dy/dx\) at \((6,8)\)?

Exercise 3

Match the implicit curve to its derivative dy/dx (drag to first empty zone):

\(x^2 + y^2 = r^2\) → Drop here
\(xy = k\) → Drop here
\(x^3 + y^3 = 1\) → Drop here
Click / tap any item to place it in the first empty zone:
-x/y
-y/x
-(3x²)/(3y²) = -x²/y²
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