2014 Derivatives of Inverse Functions AP Calculus

Monotonic – always increasing or always decreasing Inverses Existence of an Inverse: If f(x) is one-to-one on its domain D , then f is called invertible. Further, Domain of f = Range of f -1 Range of f = Domain of f -1 𝑦= 𝑥 2 𝑆𝑤𝑎𝑝 𝑥 𝑎𝑛𝑑 𝑦 𝑥= 𝑦 2 ± 𝑥 =𝑦 One-to One Functions: A function f(x) is one-to one (on its domain D) if for every x there exists only one y and for every y there exists only one x Horizontal line test.

Find the inverse 𝑦= 𝑥+3 𝑥+1 𝑥= 𝑦+3 𝑦+1 Switch x and y multiply 𝑥 𝑦+1 =𝑦+3 𝑥𝑦+𝑥=𝑦+3 distribute Collect y 𝑥𝑦−𝑦=3−𝑥 𝑦 𝑥−1 =3−𝑥 factor 𝑦= 3−𝑥 𝑥−1 divide

Find the inverse 𝑦= 3 𝑥+4 𝑥= 3 𝑦+4 𝑥 3 =𝑦+4 𝑥 3 −4=𝑦

𝑦= 𝑥 2 −4 for x ≥ 2 makes it monotonic

REVIEW: Inverse Functions If f(x) is a function and ( x, y) is a point on f(x) , then the inverse f -1(x) contains the point ( y, x) To find f -1(x) Reverse the x and y and resolve for y. (a,b) (b,a) Theorem: f and g are inverses iff f(g(x)) = g(f(x)) = x

𝑓 𝑥 = 𝑥 3 −4 𝑔 𝑥 = 3 𝑥+4 𝑓 𝑔 𝑥 =𝑔 𝑓 𝑥 3 𝑥+4 3 −4= 3 𝑥 3 −4+4 𝑥+4−4= 3 𝑥 3 𝑥=𝑥

Restricting the Domain: If a function is not one-to-one the domain can be restricted to portions that are one-to-one.

Restricting the Domain: If a function is not one-to-one the domain can be restricted to portions that are one-to-one. Increasing (−∞,−2) Decreasing (−2,3) Increasing (3,∞) Has an inverse on each interval

Find the derivative of the inverse by implicit differentiation ( without solving for f -1 (x) ) Remember : f -1 (x) = f (y) ; therefore, find 𝑦=2𝑥+sin⁡(𝑥) 𝑑𝑦 𝑑𝑦 =2 𝑑𝑥 𝑑𝑦 +cos⁡(𝑥) 𝑑𝑥 𝑑𝑦 1=(2+𝑐𝑜𝑠 𝑥 ) 𝑑𝑥 𝑑𝑦 1 2+cos⁡(𝑥) = 𝑑𝑥 𝑑𝑦

Derivative of the Inverse f(a,b) =m Derivative of the Inverse (a,b) The SLOPES of the function and its inverse at the respective points (a,b) and (b,a) are reciprocals. 𝑓 −1 𝑏,𝑎 = 1 𝑚 (b,a) f(x) slope @ a = 3 𝑓 −1 𝑥 𝑠𝑙𝑜𝑝𝑒 @𝑏= 1 3 Derivative of an Inverse Function: Given f is a differentiable one-to-one function and f -1 is the inverse of f . If b belongs to the domain of f -1 and f /(f(x) ≠ 0 , then f -1(b) exists and = 1 𝑓′(𝑎)

Derivative of the Inverse (a,b) The SLOPES of the function and its inverse at the respective points (a,b) and (b,a) are reciprocals. (b,a) Derivative of an Inverse Function: If is the derivative of f, Then is the derivative of f -1(b) CAUTION: Pay attention to the plug in value!!!

Find the derivative of f -1 at (16,4) (4,16) ILLUSTRATION: (16,4) Find the derivative of f -1 at (16,4) a) Find the Inverse. b) Use the formula. 𝑓 −1 𝑥 = (𝑥) 1 2 𝑦= 𝑥 2 𝑦 ′ =2𝑥 (𝑓 −1 )′(𝑥)= 1 2 (𝑥) −1 2 𝑦 ′ = 1 2𝑥 (𝑓 −1 )′(𝑥)= 1 2 𝑥 𝑦 ′ = 1 2(4) = 1 8 (𝑓 −1 )′(𝑥)= 1 2 16 = 1 8 𝑓 −1 ′ 𝑏 = 1 8

EX: Find the derivative of the Inverse at the given point, (b,a). 6= 𝑥 3 +7 −1= 𝑥 3 (-1,6) 𝑓 ′ 𝑥 = 3𝑥 2 𝑓 ′ −1 =3 (𝑓 −1 )′(6)= 1 𝑓 ′ −1 = 1 3 Theorem:

x f f / 10 3 2 3 10 4 Inverse Functions 𝑠 ′ 3 = 1 𝑓′(10) = 1 2 REMEMBER: The x in the inverse (S) is the y in the original (f) 𝑠 ′ 3 = 1 𝑓′(10) = 1 2 If S(x) = f -1 (x), then S / (3) = If S(x) = f -1 (x), then S / (10) = 3 is the y value 𝑠 ′ 10 = 1 𝑓′(3) = 1 4 10 is the y value

Last Update 1/8/14 Assignment: Worksheet 91