Summary of Derivative Rules
1
Summer 2016
General Derivative Rules
1. Constant Rule
d
[c ] = 0
dx
2. Constant Multiple Rule
d
[cf (x )] = cf 0 (x )
dx
3. Sum Rule
d
[f (x ) + g (x )] = f 0 (x ) + g 0 (x )
dx
4. Difference Rule
d
[f (x ) − g (x )] = f 0 (x ) − g 0 (x )
dx
5. Product Rule
6. Quotient Rule
7. Chain Rule
2
d
[f (x )g (x )] = f 0 (x )g (x ) + f (x )g 0 (x )
dx
g (x )f 0 (x ) − f (x )g 0 (x )
d f (x )
=
dx g (x )
[g (x )]2
d
[f (g (x ))] = f 0 (g (x ))g 0 (x )
dx
Derivative Rules for Particular Functions
Basic Rule
Chain Rule Form
1. Powers
d
[x n ] = nx n−1
dx
d
[(f (x ))n ] = n(f (x ))n−1 f 0 (x )
dx
2. Sine
d
[sin x ] = cos x
dx
d
[sin (f (x ))] = cos (f (x ))f 0 (x )
dx
3. Cosine
d
[cos x ] = − sin x
dx
d
[cos (f (x ))] = − sin (f (x ))f 0 (x )
dx
4. Tangent
d
[tan x ] = sec2 x
dx
d
[tan (f (x ))] = sec2 (f (x ))f 0 (x )
dx
5. Secant
d
[sec x ] = sec x tan x
dx
d
[sec (f (x ))] = sec (f (x )) tan (f (x ))f 0 (x )
dx
6. Cosecant
d
[csc x ] = − csc x cot x
dx
d
[csc (f (x ))] = − csc (f (x )) cot (f (x ))f 0 (x )
dx
7. Cotangent
d
[cot x ] = − csc2 x
dx
8. Exponential (base e )
d
[e x ] = e x
dx
9. Exponential (base a)
d
[ax ] = ax ln a
dx
d
[cot (f (x ))] = − csc2 (f (x ))f 0 (x )
dx
d h (f (x)) i
e
= e (f (x)) f 0 (x )
dx
d h (f (x)) i
a
= a(f (x)) ln af 0 (x )
dx
10. Natural Logarithm
d
1
[ln x ] =
dx
x
d
1 0
[ln f (x )] =
f (x )
dx
f (x )
11. Logarithm (base a)
d
1
[loga x ] =
dx
x ln a
d
1
0
[loga f (x )] =
f (x )
dx
f (x ) ln a
12. Inverse sine
d
1
[arcsin x ] = √
dx
1 − x2
d
1
0
[arcsin f (x )] = p
f (x )
dx
1 − (f (x ))2
13. Inverse cosine
d
−1
[arccos x ] = √
dx
1 − x2
d
−1
0
[arccos f (x )] = p
f (x )
dx
1 − (f (x ))2
14. Inverse tangent
d
1
[arctan x ] =
dx
1 + x2
d
1
0
[arctan f (x )] =
f (x )
dx
1 + (f (x ))2
p. 1 of 2
Summary of Derivative Rules
3
Summer 2016
General Antiderivative Rules
Let F (x ) be any antiderivative of f (x ). That is, F 0 (x ) = f (x ). The most general antiderivative of f (x ) is then F (x ) + C .
Original Function
General Antiderivative
1. Constant Rule
c (a constant)
cx + C
2. Constant Multiple Rule
cf (x )
cF (x ) + C
3. Sum Rule
f (x ) + g (x )
F (x ) + G (x ) + C
4. Difference Rule
f (x ) − g (x )
F (x ) − G (x ) + C
4
Antiderivative Rules for Particular Functions
Original Function
General Antiderivative
1. Powers (n 6= −1)
xn
x n+1
+C
n+1
2. Powers (n = −1)
1
x
ln |x | + C
3. Sine
sin x
− cos x + C
4. Cosine
cos x
sin x + C
5. Secant squared
sec2 x
tan x + C
6. Secant times tangent
sec x tan x
sec x + C
7. Cosecant times cotangent
csc x cot x
− csc x + C
8. Cosecant squared
csc2 x
− cot x + C
9. Exponential (base e )
ex
ex + C
10. Exponential (base a)
ax
ax
+C
ln a
p. 2 of 2