Calculus1 Lecture 19: Implicit Differentiation скачать в хорошем качестве

Calculus1 Lecture 19: Implicit Differentiation

In this lecture, we introduce implicit differentiation and explain how derivatives can be computed when a function is defined implicitly rather than explicitly. Topics covered: Explicit versus implicit functions Implicit differentiation using the chain rule Solving for dy/dx ​ Implicit differentiation involving trigonometric functions Second derivatives in implicit form Tangent and normal lines to implicitly defined curves Angles between curves Intersection angles between a line and a curve Angles of intersection between two circles Several worked examples illustrate how implicit differentiation is applied to compute derivatives, find tangent lines, and determine angles between intersecting curves. Instructor: Prof. Shadi Shaqaqha Department of Mathematics – Yarmouk University شرح الاشتقاق الضمني (Implicit Differentiation) وكيفية إيجاد المشتقة عندما تكون y معرفة ضمنيًا، مع تطبيقات على المماس والزاوية بين المنحنيات. 00:20 Explicit vs implicit functions 01:01 Implicit differentiation procedure 01:50 Example 1 – Polynomial implicit equation 04:34 Example 2 – Trigonometric implicit equation 07:05 Example 3 – Second derivative implicitly 10:01 Tangent and normal lines 13:23 Angle between a line and a curve 19:05 Angle of intersection of two circles