[Sarkari-Naukri] Harish Sati, sample questions Derivatives

Sample Questions

Derivatives

Question 1

Write down the following derivatives:

(a)

d

dx

(12

x4 − 3x3 + 7x2 − 3x + 4)

(b)

d

dx

(sin 2

x)

(c)

d

dx

(sec

x)

Question 2

Write down the following derivatives:

(a)

d

dx

(16

x4 + 12x2 − 3x−1)

(b)

d

d

(tan

cosec )

(c)

d

dt

((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 = 0

Question 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 3

1

−1 −1 5

The 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 + 1

x

− 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 x

3

x2 + 5x − 2

50

with respect to

x.

Question 8

Consider the function

f

(x) =

(

x

sin(1/x) x 6= 0

0

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)2

t

+ 1

1/2

.

Question 10

Using limit techniques, show that if

f and g are differentiable functions, then

d

dx

(

f + g) = df

dx

+

dg

dx

.

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

t

t

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 by

dy

dx

=

d

dx

(

uv) = u

dv

dx

+

v

du

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

103

dx

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 = 16

at 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)3

p

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

2

y

+

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 differentiation

easier. 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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