Honors 2202-2
Review of Basic Functions and Derivatives
Polynomials
Polynomials are sums of functions of the form axn , where a and n are real
d n
numbers. If n 6= 0 is a real number, then dx
x = nxn−1 . If n = 0, then the
d
function is a constant (call it a) and dx a = 0.
Natural Log and e ≈ 2.718281828
(1) If A > 0 and B are real numbers, then ln A = B is equivalent to A = eB .
(2) If A > 0 and B > 0 are real numbers, then ln A + ln B = ln AB and
A
.
ln A − ln B = ln B
(3) If A and B are real numbers, then eA+B = eA · eB .
d x
d
(4) dx
e = ex and dx
ln x = x1
Trigonometric Functions
The trigonometric functions, sin z and cos z are defined based on the unit
circle. For a given angle z, cos z is defined as the x-coordinate and sin z is
defined as the y-coordinate for that angle on the unit circle (see figures).
Because the x and y coordinates on the unit circle have magnitude at most
1, we know that −1 ≤ cos x ≤ 1 and −1 ≤ sin x ≤ 1. The angle z is usually
defined in radians (1800 = π radians).
sin x (see figure)
d
(1) dx
sin x = cos x
(2) sin x is an odd function - symmetric about the origin and
sin (−x) = − sin x
cos x (see figure)
d
(1) dx
cos x = − sin x
(2) cos x is an even function - symmetric over the y-axis and
cos (−x) = cos x
Product Rule
"
#
"
dg
df
d
· g(x) + f (x) ·
[f (x) · g(x)] =
dx
dx
dx
#
[f (x) · g(x)]0 = f 0 (x) · g(x) + f (x) · g 0(x)
1
Table 1: Important values of sin x and cos x (x is given in radians)
x
0
cos x 1
sin x
0
π
6
√
3
2
1
2
π
3
π
2
π
3π
2
2π
1
2
0
−1
0
1
1
0
−1
0
√
3
2
Examples:
i
d h 3
(x + 1) · (x5 − 2x2 + 2) = (3x2 ) · (x5 − 2x2 + 2) + (x3 + 1) · (5x4 − 4x)
dx
i
d h
sin x · (x2 − 1) = (cos x) · (x2 − 1) + (sin x) · (2x)
dx
Chain Rule
d
[f (g(x))] = f 0 (g(x)) · g 0 (x)
dx
d
dy du
[y(u(x))] =
·
dx
du dx
Examples:
d
5
[ln (5x + 1)] =
dx
5x + 1
100 99
d 3
x + 2x + 1
= 100 · x3 + 2x + 1 · (3x2 + 2)
dx
i
d h√
1
1
sin x = · √
· cos x
dx
2
sin x
2
Unit Circle
1
sin(z)
angle z
y
0
-1
-1
0
x
cos(z) 1
Unit Circle
1
y
sin(z)
angle z
0
0
cos(z)
1
x
Sin(x)
1
y
0.5
0
-0.5
-1
-2pi -3pi/2 -pi -pi/2
3
0
x
pi/2
pi 3pi/2 2pi
Cos(x)
1
y
0.5
0
-0.5
-1
-2pi -3pi/2 -pi -pi/2
0
x
4
pi/2
pi 3pi/2 2pi