University:
Felician University
Course:
MATH-231 | Geometry I
Academic year:

2023
Views:

200

Pages:

4
Author:

SpaceCadet101

CALCULUS AB: DERIVATIVES OF INVERSE FUNCTIONS NOTES Objectives: Find the derivative of an inverse function (polynomials, rationals, and radicals Part 1) Differentiate an inverse trigonometric function (Part 2). Part 1 Derivatives of Inverse Functions THEOREM 2.16 Continuity and Differentiability of inverse Functions Let f be a function whose domain is an interval 1. If f has an inverse function. then the following statements are true. L. If f is continuous on its domain, then is coutinuous on lis domain 2. If is differentiable on an interval containing e and fle). then f is differentiable at fic). A proof of this theorem is given in Appendis A. ka THEOREM 2.17 The Derivative of an Inverse Function Let be a function that is differentiable on an interval f. It has an inverse function g, then y is differentiable at any for which tele Moreover, A proof of this theorem is given in Appendix A. The graph of a reflection of the graph of fan the line y Figure 2.29 The reasonableness of Theorem 2:16 follows from the reflective property of inverse functions, as shown in Figure 2.29. When determining the derivative of an inverse function, you have two options: (1) you can apply Theorem 2.17, or (2) you can use implicit differentiation. Example 1: Let f(x) = 1/4x^3 + x - 1. Verify that f(x) has an inverse. Then use the function f and the given real number a = 3 to find (f^(-1))'(a). Step 1: Find the value of x when f(x) = 3 3 = 1/4x^3 + x - 1 4 = 1/4x^3 + x 16 = x^3 + 4x 16 = x(x^2 + 4) x = 2 Step 2: Write the relationship using inverses. f(x) -> (2, 3) f^(-1)(x) -> (3, 2) Step 3: Find f'(x) = 3/4x^2 + 1 Step 4: Find f'(2) = 3/4(4) + 1 = 3 + 1 = 4 Step 5: (f^(-1))'(3) will be the reciprocal of f'(2) (f^(-1))'(3) = 1/4 Example 2: Let f(x) = (x+2)/(x-1), x > 1. Verify that f(x) has an inverse. Then use the function f and the given real number a = 2 to find (f^(-1))'(a). STEP 1: FIND X WHEN F(X) = 2 2 = (x+2)/(x-1) => 2x - 2 = x + 2 => x = 4 STEP 2: WRITE INVERSES F(x) -> (4, 2) F^(-1)(x) -> (2, 4) STEP 3: FIND F'(x) = (x-1)(1) - (x+2)(1)/(x-1)^2 = -3/(x-1)^2 STEP 4: FIND F'(4) = -3/(4-1)^2 = -3/9 = -1/3 = m of F(x) AT X = 4 STEP 5: FIND (F^(-1))'(2) = RECIPROCAL = -1 = m OF F^(-1)(x) AT X = 2 Part 2 Derivatives of Inverse Trigonometric Functions The following theorem lists the derivatives of the six inverse trigonometric functions. Note that the derivatives of arccos u, arccot u, and arccsc u are the negatives of the derivatives of arcsin u, arctan u, and arcsec u, respectively. THEOREM 2.18 Derivatives of Inverse Trigonometric Functions Let u be a differentiable function of x. d/dx[arcsin u] = u' / √(1 - u^2) d/dx[arctan u] = u' / (1 + u^2) d/dx[arcsec u] = u' / |u|√(u^2 - 1) d/dx[arccos u] = -u' / √(1 - u^2) d/dx[arccot u] = -u' / (1 + u^2) d/dx[arccsc u] = -u' / |u|√(u^2 - 1) Example 4: Differentiate each a. arcsin(√x) = (1/2√x) / √(1 - x) = 1 / 2√(x(1-x)) b. arccos(x-x^2) = - (1-2x) / √(1-(x-x^2)^2) = -(1-2x) / √(1-x^2+x^4) = -(1-2x) / √(1-x^2)(1+x^2) = (-1 + 2x) / √(1-x^2)(1+x^2)