Toronto Metropolitan University
F24 MTH140 Lab 8 Problem Sheet
Derivatives of Inverse Functions and Implicit Differentiation,
Derivatives of Exponential and Logarithmic Functions
1. [(v1) 3.7 Problem 272 ] For the following function, find (f −1 )′ (a).
f (x) = x + sin(x),
a=0
dy
for the
2. [(v1)3.7 Problem 288 ] For the following exercise, find dx
given function:
y = x · csc−1 (x)
3. [(v1) 3.7 Problem 298 ] A television camera at ground level is 2000
feet away from the launching pad of a space rocket that is set to take
off vertically, as seen in the following figure. The
angle of elevation of
x
−1
, where x is the height
the camera can be found by θ = tan
2000
of the rocket. Find the rate of change of the angle of elevation after
launch when the camera and the rocket are 5000 feet apart.
1
4. [(v1) 3.8 Problem 304,308 ] For the following exercise, use implicit
dy
.
differentiation to find dx
a)
xy − cos(xy) = 1
b)
(xy)2 + 3x = y 2
5. [(v1) 3.8 Problem 318 ] Find all points on the graph of y 3 − 27y =
x2 − 90 at which the tangent line is vertical.
6. [(v1) 3.9 Problem 350 ] For the following exercises, find f ′ (x).
f (x) = 3sin(3x)
2
7. [(v1) 3.9 Problem 350 ] For the following exercises, use logarithmic
dy
.
differentiation to find dx
y = (x2 − 1)ln(x)
3