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Understanding Inverse Trigonometric Functions and Their Derivatives

This resource explores the evaluation and derivatives of inverse trigonometric functions. It includes the evaluation of arcsin(5x), arccos(-½), and tangent lines for inverse functions. Key problem statements involve finding the tangent line for the inverse function of a cubic polynomial and determining derivatives of various inverse trig functions such as arccos and arctan. The document also highlights the relationship between the derivatives of arcsin and arccos at specific points, emphasizing their interconnectedness in calculus.

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Understanding Inverse Trigonometric Functions and Their Derivatives

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  1. Warm Up Evaluate: arccos (- ½ ) Write an algebraic expression for tan (arcsin (5x)). 3) Given f(x) = x3 + 2x – 1 contains the point (1, 2). If g(x) is the inverse function of f(x), write an equation of the line tangent to g(x) at the point where x = 2.

  2. Derivatives of Inverse Trig Functions

  3. Determine the derivative arcsin (x) 4arcsin x 2x3arcsin x

  4. Determine the derivative 3arccos (x2)

  5. Determine the derivative arctan

  6. Determine the derivative arcsec ( )

  7. Determine the derivative arctan (cosx)

  8. Determine the derivative x3arccsc (5x)

  9. Determine the value of the derivative of y = cos x at x = /3. Determine the value of the derivative of y = arccos x at x = ½. How are these two values related?

  10. Write an equation of the line tangent to the graph of y = arccos (½x) at the point where x = 1.

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