322 Basic Engineering Mathematics
dy
dx
= 3(−2x
−3
) − 2(4 cos 4x) + 2(−e
−x
) +
1
x
= −
6
x 3 − 8 cos 4x −
2
e x +
1
x
Now try the following Practice Exercise
Practice Exercise 136 Standard derivatives
(answers on page 354)
1. Find the gradient of the curve
y = 2x 4 + 3x 3 − x + 4 at the points
(a) (0, 4) (b) (1, 8)
2. Differentiate with respect to x:
y =
2
x 2 + 2 ln 2x−2(cos 5x + 3 sin2x)−
2
e 3x
34.9 Successive differentiation
When a function y = f (x) is differentiated with respect
to x, the differential coefficient is written as
dy
dx
or f (x).
If the expression is differentiated again, the second differential coefficient is obtained and is written as
d 2 y
dx 2
(pronounced dee two y by dee x squared) or f (x)
(pronounced f double-dash x). By successive differentiation further higher derivatives such as
d 3 y
dx 3 and
d 4 y
dx 4
may be obtained. Thus,
if y = 5x
4
,
dy
dx
= 20x
3
,
d 2 y
dx 2 = 60x
2
,
d 3 y
dx 3 = 120x,
d 4 y
dx 4 = 120 and
d 5 y
dx 5 = 0
Problem 24. If f (x) = 4x 5 − 2x 3 + x − 3, find
f
(x)
f (x) = 4x
5
− 2x
3
+ x − 3
f
(x) = 20x
4
− 6x
2
+ 1
f
(x) = 80x
3
− 12x or 4x(20x
2
− 3)
Problem 25. Given y =
2
3
x 3 −
4
x 2 +
1
2x
−
√
x,
determine
d 2 y
dx 2
y =
2
3
x
3
−
4
x 2 +
1
2x
−
√
x
=
2
3
x
3
− 4x
−2
+
1
2
x
−1
− x
1
2
dy
dx
=
2
3
3x
2
− 4
− 2x
−3
+
1
2
− 1x
−2
−
1
2
x
−
1
2
i.e.
dy
dx
= 2x
2
+ 8x
−3
−
1
2
x
−2
−
1
2
x
−
1
2
d 2 y
dx 2 = 4x + (8)(−3x
−4
) −
1
2
− 2x
−3
−
1
2
−
1
2
x
−
3
2
= 4x − 24x
−4
+ 1x
−3
+
1
4
x
−
3
2
i.e.
d 2 y
dx 2 = 4x −
24
x 4 +
1
x 3 +
1
4
√
x 3
Now try the following Practice Exercise
Practice Exercise 137 Successive
differentiation (answers on page 354)
1. If y = 3x 4 + 2x 3 − 3x + 2, find (a)
d 2 y
dx 2
(b)
d 3 y
dx 3
2. If y = 4x 2 +
1
x
find
d 2 y
dx 2
3. (a) Given f (t ) =
2
5
t 2 −
1
t 3 +
3
t
−
√
t + 1,
determine f
(t ).
(b) Evaluate f (t ) in part (a) when t = 1.
4. If y = 3 sin2t + cos t , find
d 2 y
dx 2
5. If f (θ) = 2 ln4θ, show that f (θ) = −
2
θ 2
dy
dx
= 3(−2x
−3
) − 2(4 cos 4x) + 2(−e
−x
) +
1
x
= −
6
x 3 − 8 cos 4x −
2
e x +
1
x
Now try the following Practice Exercise
Practice Exercise 136 Standard derivatives
(answers on page 354)
1. Find the gradient of the curve
y = 2x 4 + 3x 3 − x + 4 at the points
(a) (0, 4) (b) (1, 8)
2. Differentiate with respect to x:
y =
2
x 2 + 2 ln 2x−2(cos 5x + 3 sin2x)−
2
e 3x
34.9 Successive differentiation
When a function y = f (x) is differentiated with respect
to x, the differential coefficient is written as
dy
dx
or f (x).
If the expression is differentiated again, the second differential coefficient is obtained and is written as
d 2 y
dx 2
(pronounced dee two y by dee x squared) or f (x)
(pronounced f double-dash x). By successive differentiation further higher derivatives such as
d 3 y
dx 3 and
d 4 y
dx 4
may be obtained. Thus,
if y = 5x
4
,
dy
dx
= 20x
3
,
d 2 y
dx 2 = 60x
2
,
d 3 y
dx 3 = 120x,
d 4 y
dx 4 = 120 and
d 5 y
dx 5 = 0
Problem 24. If f (x) = 4x 5 − 2x 3 + x − 3, find
f
(x)
f (x) = 4x
5
− 2x
3
+ x − 3
f
(x) = 20x
4
− 6x
2
+ 1
f
(x) = 80x
3
− 12x or 4x(20x
2
− 3)
Problem 25. Given y =
2
3
x 3 −
4
x 2 +
1
2x
−
√
x,
determine
d 2 y
dx 2
y =
2
3
x
3
−
4
x 2 +
1
2x
−
√
x
=
2
3
x
3
− 4x
−2
+
1
2
x
−1
− x
1
2
dy
dx
=
2
3
3x
2
− 4
− 2x
−3
+
1
2
− 1x
−2
−
1
2
x
−
1
2
i.e.
dy
dx
= 2x
2
+ 8x
−3
−
1
2
x
−2
−
1
2
x
−
1
2
d 2 y
dx 2 = 4x + (8)(−3x
−4
) −
1
2
− 2x
−3
−
1
2
−
1
2
x
−
3
2
= 4x − 24x
−4
+ 1x
−3
+
1
4
x
−
3
2
i.e.
d 2 y
dx 2 = 4x −
24
x 4 +
1
x 3 +
1
4
√
x 3
Now try the following Practice Exercise
Practice Exercise 137 Successive
differentiation (answers on page 354)
1. If y = 3x 4 + 2x 3 − 3x + 2, find (a)
d 2 y
dx 2
(b)
d 3 y
dx 3
2. If y = 4x 2 +
1
x
find
d 2 y
dx 2
3. (a) Given f (t ) =
2
5
t 2 −
1
t 3 +
3
t
−
√
t + 1,
determine f
(t ).
(b) Evaluate f (t ) in part (a) when t = 1.
4. If y = 3 sin2t + cos t , find
d 2 y
dx 2
5. If f (θ) = 2 ln4θ, show that f (θ) = −
2
θ 2