Math 1100
Learning Centre
Differentiation Rules
CONSTANT RULE
d
dx n = 0, where n ∈ ℝ
POWER RULE
d
n
(n − 1)
, where n ∈ ℝ, n ≠ 0
dx x = nx
The derivative of a constant is zero.
The derivative of a power of x is the old
exponent times x raised to the old power
minus one
COEFFICIENT RULE
d
d
dx n · ƒ(x) = n · dx ƒ(x)
SUM & DIFFERENCE RULES
d
d
d
dx (ƒ(x) ± g(x)) = dx ƒ(x) ± dx g(x)
The coefficient of an expression
doesn’t affect the derivative of that
expression.
The derivative of a sum (or difference)
of expressions is the sum (or difference) of
the derivatives of the individual expressions.
BASIC DERIVATIVES
d
dx
x=1
d
dx
sin x = cos x
d
dx
cos x = −sin x
d
dx
tan x = sec² x
d
dx
ex = e x
d
dx
ln x =
1
x
,x>0
Example 1: Use one of the rules to find the derivative: a) x4 b) 4x c) sin x + cos x.
Solution:
d
a) Power Rule: dx
x4 = 4x³
d
b) Coefficient Rule: dx
4x = 4 · 1 = 4
d
d
c) Sum Rule: dx (sin x + cos x) = dx
sin x +
PRODUCT RULE
d
dx
d
dx
cos x = cos x + (− sin x)
QUOTIENT RULE
d
dx
(ƒ(x) · g(x)) = ƒ′(x)g(x) + g′(x)ƒ(x)
ƒ(x)
g(x)
=
ƒ′(x)g(x) − g′(x)ƒ(x)
[g(x)]²
You will sometimes see the Product Rule written slightly differently, but remembering
that the derivative of the first part comes first (though the order doesn’t matter in the end)
will help you to remember the same pattern in the Quotient Rule (where the order does
matter).
Example 2: Use the Product Rule to find the derivative of x · ln x.
Solution:
d
dx
1
x · ln x = (1 · ln x) + (x · x ) = ln x + 1
© 2013 Vancouver Community College Learning Centre.
Student review only. May not be reproduced for classes.
AuthoredbybyEmily
Darren
Rigby
Simpson
OTHER DERIVATIVES
1
d
dx
csc x = −csc x · cot x
d
dx
sin−1 x = 1 − x²
d
dx
ax = ax ln a , x > 0
d
dx
sec x = sec x · tan x
d
dx
cos−1 x = − 1 − x²
1
d
dx
loga x = x ln a , x > 0
d
dx
cot x = −csc2x
d
dx
tan−1 x = 1 + x²
1
1
EXERCISES
Note: You should be able to do these without referring to the previous page!
A. Find the derivative:
1) ln x − x
4) 5 cos x
1
2) x
5) e
3) x17
6) ³ x
B. Use the definitions of the following functions and the Quotient Rule to prove the list
of derivatives given at the top of this page.
1) csc x
2) sec x
3) cot x
C. Find the derivative. Do not simplify the results:
1) 6x³ + 5x² − 7x + 2
5) (x³ − 8x² + x)(log x)
4
3x + 17x³ − 14x² − 73x + 63
x² + 4x − 9
2) 3 x19 − 4 x15 + 5 x11
6)
3) (x − 3)²
7) cos‾¹ x
4) sec x · ln x
8) 3x · x³ · 3x [Hint: there’s a trick.]
57
45
165
cos x
SOLUTIONS
A. (1) 1x − 1 (2) − x²1 (3) 17x16 (4) −5 sin x (5) 0 (6) 13 x−2/3 =
³x
1
3 ³ x² = 3x
−1/cos² x
sin² x/cos² x =
x
sin x
−sec² x
B. (1) −cos
csc² x
sin² x = −csc x cot x (2) cos² x = sec x tan x (3) tan² x =
18
4 14
1 10
C. (1) 18x² + 10x − 7 (2) x − 3 x + 3 x
(3) expand first; 2x − 6
sec x
(4) sec x tan x ln x + x
(5) (3x² − 16x + 1)(log x) + x² −ln8x10+ 1
− 9) − (3x4 + 17x³ − 14x² − 73x + 63)(2x + 4)
(6) (12x³ + 51x² − 28x − 73)(x² + 4x(x²
, or 6x + 5, if you divided first.
+ 4x − 9)²
(7) (−sin x cos−1 x +
cos x
1 − x²
)(cos−1 x)−2 (8) x4 · 3x + 1 · ln 3 + 4x3 · 3x + 1
© 2013 Vancouver Community College Learning Centre.
Student review only. May not be reproduced for classes.
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