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.
- سوريا: الصف ١٢
- فلسطين: الصف ١٢
- إيران: الصف ١٢
- أفغانستان: الصف ١٢
- روسيا: الصف ١٠
- ألمانيا: الصف ١١
- فرنسا: الصف ١٢
- بريطانيا: السنة ١٢
لم يُترجَم هذا المختبر إلى هذه اللغة بعد.
الدليل
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): (y٢)′ = ٢y·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 = x٢ then dAdt = ٢x·dxdt.