Ma 113 Functions And Inverse Functions, The Exponential Function And The Logarithm Worksheet Page 19
ADVERTISEMENT
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41Worksheet # 14: Implicit Differentiation and Inverse Functions
1. Find the derivative of y with respect to x:
4
(a) ln (xy) = cos(y
).
2
2
2
(b) x
+ y
= π
.
3
3
3
(c) sin(xy) = ln
.
2
2
(
2)
(
3)
2. Consider the ellipse given by the equation
+
= 1.
25
81
(a) Find the equation of the tangent line to the ellipse at the point (u, v) where u = 4 and v > 0.
(b) Sketch the ellipse and the line to check your answer.
1
tan
(
)
3. Find the derivative of f (x) = π
, where ω is a constant.
2
2
4. Let (a, b) be a point in the circle x
+ y
= 144. Use implicit differentiation to find the slope of the
tangent line to the circle at (a, b).
1
1
5. Let f (x) be an invertible function such that g(x) = f
(x), f (3) =
5 and f (3) =
. Using only
2
this information find the equation of the tangent line to g(x) at x =
5.
1
1
6. Let y = f (x) be the unique function satisfying
+
= 4. Find the slope of the tangent line to f (x)
2x
3y
1
1
at the point (
,
).
2
9
7. The equation of the tangent line to f (x) at the point (2, f (2)) is given by the equation y =
3x + 9.
x
If G(x) =
, find G (2).
4f (x)
4
dr
3
8. Differentiate both sides of the equation, V =
πr
, with respect to V and find
when r = 8 π.
3
dV
9. Use implicit differentiation to find the derivative of arctan(x). Thus if x = tan(y), use implicit differ-
entiation to compute dy/dx. Can you simplify to express dy/dx in terms of x?
d
10. (a) Compute
arcsin(cos(x)).
dx
(b) Compute
(arcsin(x) + arccos(x)). Give a geometric explanation as to why the answer is 0.
d
1
1
1
(c) Compute
tan
+ tan
(x)
and simplify to show that the derivative is 0. Give a
dx
x
geometric explanation of your result.
11. Consider the line through (0, b) and (2, 0). Let θ be the directed angle from the x-axis to this line so
that θ > 0 when b < 0. Find the derivative of θ with respect to b.
2
12. Let f be defined by f (x) = e
.
(a) For which values of x is f (x) = 0
(b) For which values of x is f (x) = 0
1
13. The notation tan
(x) is ambiguous. It is not clear if the exponent
1 indicates the reciprocal or
the inverse function. If we allow both interpretations, how many different ways can you (correctly)
compute the derivative f (x) for
1
1
f (x) = (tan
)
(x)?
In order to avoid this ambiguity, we will generally use cot(x) for the reciprocal of tan(x) and arctan(x)
for the inverse of the tangent function restricted to the domain ( π/2, π/2).
ADVERTISEMENT
0 votes
Related Articles
Related forms
Related Categories
Parent category: Education