Implicit, parametric and second derivatives, and connected rates
Find the gradient of a circle at a point by implicit differentiation, work out the gradient of a parametric curve, take a second derivative, and see how a changing side makes the area change.
- UK: Year 12
- France: grade 12
- Germany: grade 11
- Iran: grade 12
- Afghanistan: grade 12
- Syria: grade 12
- Palestine: grade 12
- Russia: grade 10
Guide
If y is tied to x by an equation but not isolated, differentiate both sides with respect to x and attach a factor y′ wherever y appears (the chain rule): (y2)′ = 2y·y′.
For a parametric curve x(t), y(t): dydx = dydtdxdt. The second derivative is the derivative of the derivative; it shows how the gradient itself changes.
When two quantities are tied together, so are their rates of change: if A = x2 then dAdt = 2x·dxdt.