Download Differentiation - Inverse Trigonometric Functions

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Kuta Software - Infinite Calculus
Differentiation - Inverse Trigonometric Functions
1) y = cos −1 −5x 3
2) y = sin −1 −2x 2
3) y = tan −1 2x 4
5) y = (sin −1 5x 2 )
3
7) y = (cos −1 4x 2 )
6) y = sin −1 (3x 5 + 1) 3
2
8) y = cos −1 (−2x 3 − 3) 3
Derivatives of inverse function – PROBLEMS
𝑓𝑓 (𝑔𝑔 (𝑥𝑥 )) = 𝑥𝑥
.
𝑓𝑓′(𝑔𝑔(𝑥𝑥))𝑔𝑔′(𝑥𝑥) = 1
𝑔𝑔′ (𝑥𝑥) =
1
𝑓𝑓′(𝑔𝑔(𝑥𝑥))
The beauty of this formula is that we don’t need to actually determine 𝑔𝑔(𝑥𝑥) to find the value of
the derivative at a point. We simply use the reflection property of inverse function:
Derivative of the inverse function at a point is the reciprocal of the derivative of the
function at the corresponding point.
Slope of the line tangent to 𝒇𝒇−𝟏 at 𝒙𝒙 = 𝒃𝒃 is the reciprocal of the slope of 𝒇𝒇 at 𝒙𝒙 = 𝒂𝒂.
1. Find tangent line at point (4, 2) of the graph of f -1 if f(x) = x3 + 2x – 8
2. Find the equation of the tangent line to the inverse at the given point.
a. f(x) = x3 + 7x +2
@ (10, 1)
b. f(x) = x5 + 3x3 + 7x +2
@ (13, 1)
c. f(x) = e-2x – 9x3 + 4
d. f(x) = x7 + 2x + 9
e. f(x) = x5/3 𝑒 𝑥
𝑓𝑓.
𝑓𝑓(𝑥𝑥) =
2
−𝑒 −3𝑥
𝑥𝑥 2 + 1
g. f(x) = 7x + sin (2x)
@ (5, 0)
@ (12, 1)
@ (e, 1)
@ (−1, 0)
@ (0,0)
2
h. f(x) = x3 + 8x + cos (3x)
@ (1,0)
i. f(x) = 10x + (arc tanx)2
@ (0,0)
j. f(x) = 7x3 + (ln x)3
@ (7, 1)
3. A function 𝑓𝑓 and its derivative take on the values shown in the table. If 𝑔𝑔 is the inverse of 𝑓𝑓,
find 𝑔𝑔′ (6).
x
2
6
f(x)
6
8
f’(x)
1/3
3/2
4. Let y = f(x) = x3 + x + x – 2, and let g be the inverse function. Evaluate g’(0).