Definition: Implicit differentiation finds dy/dx for a relation where y is not isolated, by differentiating both sides with respect to x and treating y as a function of x.
When do you use it?
Use it when an equation mixes x and y and solving for y is messy or impossible, such as x2 + xy + y2 = 7. Differentiate every term with respect to x. Whenever you differentiate a y-term, attach dy/dx by the chain rule.
How does it work?
Worked example: the point (1, 2) lies on x2 + xy + y2 = 7 since 1 + 2 + 4 = 7. Differentiate: 2x + y + x(dy/dx) + 2y(dy/dx) = 0. Notice the product rule on xy.
Group the dy/dx terms: dy/dx = -(2x + y)/(x + 2y).
At (1, 2): dy/dx = -(2 + 2)/(1 + 4) = -4/5.
The tangent line there is y – 2 = -0.8(x – 1).
What mistakes occur?
Forgetting dy/dx on y-terms, or missing the product rule on xy. Mathbot checks every y with a small tag that says “times dy/dx”.