Implicit Differentiation

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Presentation transcript:

Implicit Differentiation

This is not a function, but it would still be nice to be able to find the slope. Note use of chain rule.

This can’t be solved for y. This technique is called implicit differentiation. 1 Differentiate both sides w.r.t. x. 2 Solve for .

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

Find the equations of the lines tangent and normal to the curve at .

Examples: x2y3 – xy = 10 x2+3xy+y2 = -1 x2+y2 = 25 y3-x2 = x + y 3y + ln y = 4ex x2y= y3x + 5y + x 25x2 + 8x – 16y2 - 4y – 9 = 0

More examples: Find dy/dx y = sin x + cos y x2y + y3 + 3 = xy xy3 + xy = 6 ex + cos y = ln y6 x2 + 2xy + y4 = 1 x2y2 – ex + 2y = 4 sin x/y = ½ cos (x+y) + sin (x+y) = 1/3 ecos x + esin y = 1/4