Derivatives of Inverse Functions (including ln(x) and inverse trig functions) | Calculus Problems

The fact that f(f^-1(x))=x for all x in the domain of the inverse function f^-1 allows us to find the derivative of f^-1 by using the chain rule. This is done for the natural logarithm and for inverse trigonometric functions (sin^-1x, cos^-1x, tan^-1x, that is, arcsin, arccos, and arctan). Infinite Powers, How Calculus Reveals the Secrets of the Universe: https://amzn.to/37PBMjb.

Calculus problems and solutions

🔴 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

#calculus #inversefunction #derivatives

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(0:00) Derivative of ln(x) (the natural log function is the inverse function of e^x, use the Chain Rule)
(2:24) Derivative of ln(-x) and ln|x|
(4:06) Derivative of sin^-1(x)=arcsin(x) (arcsine is the inverse function of sin(x) when the domain of sine is restricted to -pi/2 to pi/2)
(11:03) Derivative of tan^-1(x)=arctan(x) (arctangent is the inverse function of tan(x) when the domain of tangent is restricted to -pi/2 to pi/2)
(14:07) Derivative of cos^-1(x)=arccos(x) (arccosine is the inverse function of cos(x) when the domain of cosine is restricted to 0 to pi)
(15:46) cos^-1(x)+sin^-1(x) is constant
(18:11) Derivative of ln(x^2) in two ways (one of which uses a property of exponents)
(19:36) Derivative of arctan(e^(2x)*sin(3x)) (Chain Rule and Product Rule)
(21:29) Derivative of sin^-1(x)/cos^-1(x) (Quotient Rule)
(23:45) Derivative of general inverse function f^-1(x) for an invertible function f(x) (Chain Rule)
(26:23) Graphical derivative of inverse function
(30:18) Inverse function of a cubic polynomial function (it is one-to-one, but we don’t find a formula for the inverse function)
(35:11) Population word problem (inverse function used)
(41:27) Vehicle registration word problem (inverse function used)

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