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Unformatted text preview: 3.6 Implicit Differentiation
Suppose we have equations like x2 + y 2  25 = 0, x3 + y 2  9xy = 0, y2  x = 0
dy ? dx We say that x, y are implicitly defined. But, how would we find Example 1
Find dy/dx if y 2 = x. 1. Solve for y and then differentiate. 2. Use Implicit Differentiation 1 Implicit Differentiation
1. Differentiate 2. Collect 3. Solve Examples 1. Find dy/dx when y 2  2x = 1  2y 2 2. Find dy/dx when ex 2y = 2x + 2y 3 3. Find the slope of (x2 + y 2 )2 = (x  y)2 at (1, 0). 4 Normal Lines
A line which is perpendicular to the tangent line of a curve at a point (x, y) on the curve is called normal to the curve or normal line. Recall: How are the slopes of perpendicular lines related? If the slope of a line is m, the slope of the perpendicular to this line will be Examples 1. Verify that (1, /2) is on the curve 2xy + sin y = 2 and find the tangent and normal lines at this point. 5 2. Find the two points where the curve x2 + xy + y 2 = 7 crosses the xaxis, and show that the tangents to the curve at these points are parallel. Find the common slope. 6 3. The line that is normal to the curve x2 + 2xy  3y 2 = 0 at (1, 1) intersects the curve at what other point? 7 Derivatives of Higher Order
We can also use implicit differentiation to find higherordered derivatives. Example Find d2 y/dx2 of y 2  2x = 1  2y 8 ...
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 Spring '08
 FBHinkelmann
 Calculus, Equations, Implicit Differentiation, Slope

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