# MATH 110, Assignment 4

```MATH 110, Assignment 4
Make sure you justify all your work and include complete arguments and explanations. Please include your name and student number on the top of the first
Problem 1. Differentiate.
(a) y =
ex
1+x
(b) y = eu (cos u + u)
(c) f (x) = ex cos x
Problem 2. The functions f and g are given as followings:
Find the following values:
(a) g 0 (1)
(b) (f g)0 (3)
(c) (f −1 )0 (3)
Problem 3. How many tangent lines to the curve y =
x
x+1
pass through the point (1, 2).
Problem 4. On what interval is the function f (x) = x3 ex increasing?
Problem 5. Using product rule, prove that
d
(f (x))3 = 3(f (x))2 f 0 (x).
dx
(Hint: Show that (f gh)0 = f 0 gh + f g 0 h + f gh0 first and take f = g = h.)
1
Problem 6. A ladder 10m long rests against a vertical wall. Let θ be the angle between
the top of the ladder and the wall and let x be the distance from the bottom of the ladder
to the wall. If the bottom of the ladder slides away from the wall, how fast does x changes
with respect to θ when θ = π4 ?
2
```