MAS110 Problems for Chapter 6: Differentiation
f (x + h) − f (x)
to calculate the derivative when:
h→0
h
1. Use the definition f 0 (x) = lim
(b) f (x) := x2 − x + 1
(c) f (x) := x3
1
1
(e) f (x) := √
(f ) f (x) := 2
x +1
x
√
√
√
√
[Hint for (d), (e): use a − b = (a − b)/( a + b) at the right place to calculate the limit.]
(a) f (x) := 2x − 1
√
(d) f (x) := 2x
2. Prove, from first principles, the formula for the derivative of cos(x). You may assume any limit
formulas proved in lectures.
3. Determine the constants b and c given that
(
x2 + bx + c, if x > 0,
f (x) =
0,
if x ≤ 0
is differentiable at x = 0. [Hint: compare left and right derivatives; use continuity.]
4. For x1 > 0, consider the point P = (x1 , x21 ) on the parabola y = x2 . Let L be the line through P
and the point (ν, 0) on the x-axis. Find the equation of L (and draw a picture, in a case where L is
not tangent to the parabola). Obtain a quadratic equation in x whose roots are the x-coordinates
of the points of intersection of L with the parabola. (One such root is x1 . Let’s call the other
one x2 .) What is the sum of the roots, x1 + x2 ? Deduce that if L is tangent, so that x1 = x2 ,
then ν = x1 /2 (Theorem of Appollonius). This is close in spirit to Fermat’s method for finding
tangents.
5. To four decimal places, the points
(2, 7.3891),
(2.2, 9.0250),
(2.1, 8.1662),
(2.01, 7.4633),
(2.003, 7.4113)
lie on the graph y = f (x) of a certain function. Work out the slopes of the chords from the first
point to each of the others. What do you think f 0 (2) might be? Do you recognise the function
f?
dy
from first
6. (a) For a positive integer m, let y = 1/xm . Prove the expected formula for dx
principles.
1
(b) Writing u/v = u · , deduce the Quotient Rule from the Product Rule and the Chain Rule.
v
(You may use what you proved in (a).)
7. (Yet another way to prove the Quotient Rule.) If y = u/v, write u = vy and hence, by differentiating both sides with respect to x, deduce the Quotient Rule from the Product Rule.
1
8. Starting from dx
= 1, prove by induction that
dx
Product Rule.
9. Find
d
(xn )
dx
= nxn−1 for all n ∈ N, assuming only the
dy
when
dx
3
x4
(a) y =
−3
4
2
sin x
(d) y =
1 + cos x
(g) y = ((x2 + 1)7 + 3)5
10. Find
(b) y = sin(cos x)
(c) y = (sin−1 x)5
q
√
(e) y = 1 − x
(f ) y =
(h) y =
p
sin(x2 )
1 + x2
1 + x3
dy
when
dx
(a) x2 y + xy 2 = 1
(b) x3 + y 3 = 3xy
(c) y 2 sin y = 1 + xy
dy
= (1 + ln x)xx . [Hint: take logarithms first and then differentiate.]
dx
dy
y2
then
=
. [Hint: use y = xy and then proceed as before.]
dx
x(1 − y ln x)
11. Show that if y = xx then
Show that if y = x
xx
.
..
12. Calculate the following limits using l’Hopital’s Rule.
(a)
(d)
x3 − 3x2 + 4
lim
x→2 x3 − 2x2 − 4x + 8
lim x ln x
x→0+
(b)
(e)
1 − cos x
lim
2
x→0
x
1
1
lim
−
x→0+
x sin x
2
(c)
(f )
1 − ex
lim
x→0 x − x2
lim (csc x − cot x + cos x)
x→0+