4-3 inviting inverses of trig deriv’s Derivatives of inverse trig functions

Some things to note: The inverse of a function can be found by reflecting over the line y = x (interchanging the x & y) If there is a tangent line at (a, f (a)), then the tangent line at the inverse (f (a), a) has a slope that is the reciprocal of the original tangent slope. Original Inverse (x, y) (y, x) tangent slope tangent slope

Thm: If f is differentiable at every point of an interval I, and is never 0 on I, then f has an inverse and f –1 is differentiable at every point of the interval f ( I ). (Note: The derivatives of the inverse trig functions are found by doing implicit differentiation and using trig identities.) Derivative of the Arcsine:

Derivative of the Arctangent: Ex 2) A particle moves along the x-axis so that its position at any time t  0 is . What is the velocity of the particle when t = 16?

Derivative of the Arcsecant: Ex 3) Find

Remember: Note: Derivatives of inverse cofunctions are the negatives of the derivatives of the corresponding inverse functions. Derivatives of the Arccosine, Arccotangent and Arccosecant

Remember calculator conversions: Ex 4) Find an equation for the tangent line to the graph of y = cot –1 x at x = –1. point  m

homework Pg. 167 #6 – 42 (mult of 6) Pg. 175 #3 – 27 (mult of 3)