Implicit Differentiation Example Problems and Solutions (including a Second Derivative)

To get the idea of how implicit differentiation in calculus works, it is best to start with a simple example like the unit circle x^2+y^2=1 where we can solve for y as an explicit function. This can also be solved by replacing y with f(x) or, as is usually the case, using Leibniz notation dy/dx or just y’. From there, a cubic curve is differentiated and approximated, as well as a curve involving the sine function. Finally, a circle not centered at the origin is considered by completing the square as well. Infinite Powers, How Calculus Reveals the Secrets of the Universe: https://amzn.to/37PBMjb.

🔴 Calculus 1, Lesson 1.1: Linear Functions (3Blue1Brown SoME2): https://www.youtube.com/watch?v=lmCjnxzIals

🔴 Calculus 1, Lesson 1.2: Exponential Functions: https://www.youtube.com/watch?v=47ZqW19AjD4

🔴 Calculus 1, Lesson 1.3: Quadratic Functions, Transformations of Functions, and Function Composition: https://www.youtube.com/watch?v=1T58CnSME4g

🔴 Calculus 1, Lesson 1.4: Logarithmic Functions (Intro to Horizontal Compressions & Inverse Functions): https://www.youtube.com/watch?v=oS-gvIoJf60

🔴 Calculus 1, Lesson 1.5: Trigonometric Functions and Inverse Trigonometric Functions: https://www.youtube.com/watch?v=UFYjlnGUenE

🔴 AP Calculus AB Exam Review: Practice Exam Problems & Solutions (Multiple Choice, No Calculator): https://www.youtube.com/watch?v=2AG_Dt3x7q0

🔴 AP Calculus AB Exam Review: Free Response Practice Exam Problems & Solutions: https://www.youtube.com/watch?v=RLgUwP44V_M

🔴 AP Calculus BC Exam Review: Practice Exam (30 Question Multiple Choice, No Calculator): https://www.youtube.com/watch?v=1Q7ALcwT97A

🔴 AP Calculus BC Exam Review: Free Response Practice Exam Problems & Solutions: https://www.youtube.com/watch?v=ZWisJEDP44w

🔴 Calculus 1 Exam 1 Review Problems and Solutions (Precalculus, Limits, Continuity, Derivatives): https://www.youtube.com/watch?v=UjmvUwvd7c4

🔴 Calculus 1 Course, Lecture 1: The Big Ideas (Rates & Areas, the Infinity Principle & Circular Area): https://www.youtube.com/watch?v=2V_ehJC5QUY

🔴 Calculus 1 Course Lecture Playlist: https://www.youtube.com/playlist?list=PLmU0FIlJY-MkFrKtM0S0NLu-TxZa5jBIx

#implicitfunction #implicit_differentiation #implicitdifferentiation

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(0:00) Find y’=dy/dx for x^2+y^2=1 by solving for y=f(x) first
(4:24) Find f'(x) for x^2+y^2=1 with implicit differentiation
(8:41) Find y’=dy/dx for x^2+y^2=1 with implicit differentiation
(10:47) Find y” (second derivative) for x^2+y^2=1 with implicit differentiation
(14:37) Cubic curve implicit differentiation example
(22:51) Linear approximation to cubic curve near the point
(24:46) sin(xy)=x implicit differentiation example
(31:39) Circle example (with Completing the Square first)

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