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Unformatted text preview: Section 3.5: Implicit Diﬀerentiation
Instructor: Ms. Hoa Nguyen ([email protected]) Implicit function
The relation between x and y is not explicitly expressed as y = f (x).
For examples, x2 + y 2 = 25 (a circle) or x3 + y 3 = 6xy (the folium of Descartes). The method of implicit diﬀerentiation
This consists of diﬀerentiating both sides of the implicit function with respect to x and then
solving the resulting equation for y .
Example 1 of Section 3.5 (textbook):
Example 2 of Section 3.5 (textbook):
Example 3 of Section 3.5 (textbook):
Example 4 of Section 3.5 (textbook):
1 Inverse Trigonometric Functions
If a function f is onetoone (i.e., passes the horizontal line test), then it has the
inverse function f −1 . f −1 is also onetoone and has its inverse f .
For examples, the function f (x) = sin x, −2Π ≤ x ≤ Π has the inverse f −1 (x) = sin−1 x =
2
arcsin x. Remarks:
1) The graphs of f and f −1 are reﬂected about the line y = x.
2) Recall the deﬁnition of the arcsine function: y = sin−1 x ⇔ sin y = x and − π ≤ x ≤ π .
2
2
Inverse Trigonometric Functions:
−Π
2 • f (x) = sin x, Π
2 ≤x≤ ⇔ f −1 (x) = sin−1 x = arcsin x. • f (x) = cos x, 0 ≤ x ≤ Π ⇔ f −1 (x) = cos−1 x = arccos x.
• f (x) = tan x, −Π
2 ≤x≤ Π
2 ⇔ f −1 (x) = tan−1 x = arctan x. • f (x) = cot x, 0 ≤ x ≤ Π ⇔ f −1 (x) = cot−1 x.
• f (x) = sec x, x ∈ [0, Π ) ∪ [Π, 3Π ) ⇔ f −1 (x) = sec−1 x, x ≥ 1.
2
2
• f (x) = csc x, x ∈ (0, Π ] ∪ (Π, 3Π ] ⇔ f −1 (x) = csc−1 x, x ≥ 1.
2
2
Derivatives of Inverse Trigonometric Functions
• (sin−1 x) = √1
1 − x2 • (tan−1 x) = 1
1+x2 • (sec−1 x) = √1
x x2 − 1 (cos−1 x) = − √11 x2
−
1
(cot−1 x) = − 1+x2
1
(csc−1 x) = − x√x2 −1 Example 5 of Section 3.5 (textbook): 2 ...
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This note was uploaded on 05/23/2011 for the course MAC 2311 taught by Professor Noohi during the Fall '08 term at FSU.
 Fall '08
 Noohi
 Calculus

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