Applied Mathematics - List 4 Prepared by dr in˙z. Anna Strzelewicz

Applied Mathematics - List 4 Prepared by dr in˙z. Anna Strzelewicz

Applied Mathematics - List 4
Prepared by dr inż. Anna Strzelewicz
Silesian University of Technology
Faculty of Chemistry
Definition 1 (Derivative) The derivative of the function f (x) with respect to the variable x is the function
f 0 whose value at x is
f (x + ∆x) − f (x)
∆x→0
∆x
f 0 (x) = lim
provided the limit exists.
Notation
There are many ways to denote the derivative of a function y = f (x). The most common notations are these:
df
dy
(read: dydx or the derivative of y with respect to x ) , dx
(read: the derivative of
f 0 (x), y 0 (read: y prime), dx
d
f with respect to x ), dx f (x) (read: ddx of f(x))
1. Find the average rate of change of the function over the given interval or intervals.
a) f (x) = x3 + 1
√
b) R(θ) = 4θ + 1
i)
c) h(t) = cot t
i)
d) q(t) = 2 + cos t
i) [0, π]
i)
[2,3]
ii)
[-1,1]
ii)
[ π6 , π2 ]
ii)
[−π, π]
[0,2]
[ π4 , 3π
4 ]
2. Use the definition to find the function’s derivative at the indicated point:
a) f (x) = x3
b) f (x) =
x0 = −1
1
x
x0 = 1
√
c) f (x) = 3x − 2 x x1 = 1
x2 = 4
d) f (x) = 2x2 + 2x − 5
√
e) f (x) = x x0 = 4
x0 = 2
f) f (x) = |x|
x2 = 0
x1 = 1
3. Use the definition to find the function’s derivative:
a) f (x) = 4 − x2
d) f (x) =
b) f (x) = sin(ax)
α−β
sin α − sin β = 2 cos α+β
2 sin 2
e) f (x) =
c) f (x) = sin x2
f)f (x) =
1
x
q
1
2x
+1
2
3x−1
4. Differentiate the following functions:
x4
4
x3
3
x2
2
a) f (x) = 11x5 − 6x3 + 8
b)f (x) =
d) f (x) = (x2 − 1)(x − 3)
e) f (x) = (3x − 9)(x2 + 18)
g) f (x) =
2x2 +1
x+2
j) f (x) = (1 + x1 )(1 +
m) f (x) =
h) f (x) =
1
x2 )
cos x−sin x
cos x+sin x
−
+
√
3
c) f (x) = x x2
−x
f) f (x) = (x − 1)3 (x + 1)2
1+x4
x2
i) f (x) =
√
3x−2
√ x+1
x
x
l) f (x) = ( 1+x
)( 2−x
3 )
k) f (x) = ( x−x√x )
n) f (x) = 4x5 ex + 3 tan x
o) f (x) =
tan
√ x
x
5. Differentiate composite functions:
a) f (x) = 4 sin( x2 )
√
d) f (x) = 4 2x2 + 1
g) f (x) =
√
√
1 + sin 2x − 1 − sin 2x
√
j) f (x) = arc tan tan2 x + C
b) f (x) = sinh x
√
e) f (x) = 3 2 + 3x
c) f (x) = xe2x lnx
h) f (x) = ( x3 + x2 + x)−1
f) f (x) = (x5 − x10 )20
−1
−2
i) f (x) = 7x − x−1
+ 3x2
k) f (x) = ln(ln(lnx))
l) f (x) = ln2 arc sin
3
1
2
√
x
m) f (x) =
sin2 x
cos7 x
p) f (x) = xx ,
−
2
5 cos5 x
x>0
n) f (x) = cos2
2
q) f (x) = xx ,
q
1
x
o) f (x) = 4x +
x>0
6. Find the second derivative:
a) f (x) = 7x3 − 6x5
c) f (x) = x2 −
b) f (x) = 2x5 − 6x4 + 2x − 1
d) f (x) =
7. Find the indicated derivatives:
a)
d3 1 3
dx3 ( 3 x
+ 12 x2 + x + 1)
d5
4
dx5 (ax
+ bx3 + cx2 + dx + e)
4 d4
c) dx
1 − 2x
4
b)
2
x2
x2 +a2
1
x2
√
tan x + x2 +
r)f (x) = (sin x)cos x ,
p
tan π4
0<x<
π
2