Sample Questions
Derivatives
Question 1
Write down the following derivatives:
(a)
ddx
(12
x4 − 3x3 + 7x2 − 3x + 4)(b)
ddx
(sin 2
x)(c)
ddx
(sec
x)Question 2
Write down the following derivatives:
(a)
ddx
(16
x4 + 12x2 − 3x−1)(b)
dd
(tan
cosec )(c)
ddt
((2
t3 + 1)(7 − 3t))Question 3
Suppose that functions
f(x) and g(x) and their first derivatives have the following values at 0 and 1.x f
(x) g(x) f0(x) g0(x)0 1 1 5 1/3
1 3 -4 -1/3 -8/3
Write down the first derivatives of the following combinations at the given vaules of
x:(a) 5
f(x) + g(x), x = 1(b)
f(x)g(x), x = 0(c)
f(g(x)), x = 0Question 4
Consider the following table of values for
f(x), g(x) and their derivatives:x f
(x) g(x) f0(x) g0(x)−
1 0 1 −2 31
−1 −1 5The derivative with respect to
x of y = f(g(x)) at x = 1 is:(a) 5
(b) 15
(c) 0
(d)
−2(e) None of the above.
1
Question 5
Consider the function
f(x) =p
x
+ 1. Using limit techniques, find the derivative of the function at x = 1.Verify your answer using the rules of differentiation.
Find the tangent and normal to the graph of the function at this point.
Question 6
Consider the function
g
(x) = x + 1x
− 1(a) Use the limit definition to find the derivative of
g(x).(b) Verify your answer using the quotient rule.
(c) For which
x is the slope of the tangent to the graph of g(x) equal to −1.Question 7
Differentiate
y
=p
7
x + 5 cos x3
x2 + 5x − 250
with respect to
x.Question 8
Consider the function
f
(x) =(
x
sin(1/x) x 6= 00
x = 0.(a) Use the rules of differentiation to find the derivative of
f(x) for x 6= 0.(b) Use the limit definition of the derivative to show that
f(x) is not differentiable at x = 0.(c) Use the sandwich theorem to show that
f(x) is continuous at x = 0.Question 9
Find the derivative of
h
(t) =
(
t − 1)2t
+ 11/2
.
Question 10
Using limit techniques, show that if
f and g are differentiable functions, thend
dx
(
f + g) = dfdx
+
dgdx
.
Question 11
The upper half of a circle or radius 1 is given by the equation
y =p
1
− x2. Using limit techniques, find the derivative.Find the equation of a tangent through a general point (
c,p
1
− c2) on the semicircle.Find the tangent line to the semicircle which passes through the point (
p
2
, 0).2
Question 12
The position of an object is given by the formula
s
(t) =4
tt
2 + 1.Find the formula of the velocity of the object.
Find the velocity at
t = 0 and t = 1.Question 13
The product rule says that the first derivative of
y = uv is given bydy
dx
=
ddx
(
uv) = udv
dx
+
vdu
dx
.
Use the product rule to find a formula for the
second derivative of y.Use this formula to find the second derivative of
y = (x2 + 1) cos x.Question 14
Find
d
103dx
103 (cos(2x)) .Question 15
The curve
2(
x2 + y2)2 = 25(x2 − y2)is called a
lemniscate. Find the equation of the tangent to the curve at the point (3, 1).Question 16
Find the tangent to the hyperbola
25
x2 − 9y2 = 16at the point (1
, 1).Question 17
Find the derivative of
y
= x1/x.Question 18
Use logarithmic differentiation to find the derivative of
y
=(
x3 + 10x − ln x)3p
sin
x − 2x(tan
−1 x + 1)3/2 .3
Question 19
Find the derivative of
f
(x) = cosh(ex(x + 1)).Question 20
Find the derivative of
x
ln x − x.Question 21
Two students, Bruce and Sheila, were asked to implicitly differentiate
x
2y
+
y = 3.(a) Bruce worked the problem directly, and obtained the answer
2xy(
x2−y2) . Check Bruce’s working.(b) Sheila realized that multiplying both sides of the equation by
y to get x2 + y2 = 3y made the differentiationeasier. However the answer she got was
2x(3
−2y) . Check her working.(c) Did one of the students make a mistake, or are they both correct? If you think there was a mistake, explain
what it was; if you think they are both correct, explain why the two answers are equal.
--
with warm regards
Harish Sati
Indira Gandhi National Open University (IGNOU)
Maidan Garhi, New Delhi-110068
(M) + 91 - 9990646343 | (E-mail) Harish.sati@gmail.com
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