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B1.6 – Derivatives of Inverse Trig Functions

B1.6 – Derivatives of Inverse Trig Functions. IB Math HL - Santowski. (A) Graphs of Inverse Trig Functions. The graphs of the inverse trig functions are as follows:. (B) Inverse Trig as Functions – Restrictions.

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B1.6 – Derivatives of Inverse Trig Functions

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  1. B1.6 – Derivatives of Inverse Trig Functions IB Math HL - Santowski

  2. (A) Graphs of Inverse Trig Functions The graphs of the inverse trig functions are as follows:

  3. (B) Inverse Trig as Functions – Restrictions • From the graphs previously shown, the inverse trig “relations” are not functions since the domain elements do not “match” the range elements i.e.  not one-to-one • So we need to make domain restrictions in the original function such that when we “invert”, our inverse does turn out to be a function • What domain restrictions shall we make??

  4. (B) Inverse Trig as Functions – Restrictions For y = sin(x) between a max and min (-/2 and /2) For y = cos(x)  between a max and min (0 and ) For y = tan(x)  use one cycle, say between -/2 and /2

  5. If y = sin(x), then to make the inverse, x = sin(y) and we can use implicit differentiation to find dy/dx But can we make a substitution for cos(y)?? (C) Derivative of f(x) = sin-1(x) on (-½,½)

  6. If y = cos(x), then to make the inverse, x = cos(y) and we can use implicit differentiation to find dy/dx But can we make a substitution for sin(y)?? (D) Derivative of f(x) = cos-1(x) on (0,)

  7. If y = tan(x), then to make the inverse, x = tan(y) and we can use implicit differentiation to find dy/dx But can we make a substitution for sec2(y)?? (E) Derivative of f(x) = tan-1(x) on (-½,½)

  8. (F) Summary of Trig Inverse Derivatives • The three derivatives of the inverse of the trig. primary functions are:

  9. (G) Internet Links • Calculus I (Math 2413) - Derivatives - Derivatives of Inverse Trig Functions from Paul Dawkins • Visual Calculus - Derivatives of Inverse Trig Functions • PinkMonkey.com Calculus Study Guide - Section 4.11 Derivatives of Inverse Trigonometric Function

  10. (H) Examples • Problems and Solutions to Differentiation of Inverse Trigonometric Functions from UC Davis • Differentiate y = sin-1(1-x2) • Differentiate • Differentiate y = cos-1(sin(x)) • If y = tan-1(x/y), find dy/dx

  11. (I) Homework • Stewart, 1989, Chap 7.6, p339, Q1-4

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