Implicit differentiation
When y is tangled up with x in an equation, differentiate both sides with respect to x and treat y as a function of x, using the chain rule.
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Analogy
Adjusting a seesaw: change one side and the other must shift to stay balanced.
Example
Differentiate: 2x + 2y (dy/dx) = 0.So dy/dx = -x / y.At (3, 4): slope = -3/4.Check: 9 + 16 = 25.
In practice
Gives slopes of circles and other curves that are not graphs of y = f(x).
Practice
Flashcards
- What is the slope of x^2 + y^2 = 25 at (3, 4)?-3/4, from dy/dx = -x/y.
Questions people ask
Why does y get a dy/dx when I differentiate implicitly?
Because y depends on x, so it is a function inside the equation. Differentiating y squared gives 2y times dy/dx, by the chain rule. Terms with only x differentiate as normal. Then collect the dy/dx terms and solve for it.
Can the slope formula contain both x and y?
Yes. For the circle x squared plus y squared equals 25, the slope is minus x over y, which depends on both. You need a point on the curve, such as (3, 4), to get a number: here the slope is -3/4.