Differentiation formulas
Procedural rules
Name of rule
Formula
Constant multiple
d
d
u
(
c
f
)
=
c
d
f
d
u
\frac{d}{du} (cf) = c \frac{df}{du}
d
u
d
(
c
f
)
=
c
d
u
df
Sum
d
d
u
(
f
+
g
)
=
d
f
d
u
+
d
g
d
u
\frac{d}{du} (f + g) = \frac{df}{du} + \frac{dg}{du}
d
u
d
(
f
+
g
)
=
d
u
df
+
d
u
d
g
Difference
d
d
u
(
f
−
g
)
=
d
f
d
u
−
d
g
d
u
\frac{d}{du} (f - g) = \frac{df}{du} - \frac{dg}{du}
d
u
d
(
f
−
g
)
=
d
u
df
−
d
u
d
g
Linearity
d
d
u
(
a
f
+
b
g
)
=
a
d
f
d
u
+
b
d
g
d
u
\frac{d}{du} (af + bg) = a\frac{df}{du} + b\frac{dg}{du}
d
u
d
(
a
f
+
b
g
)
=
a
d
u
df
+
b
d
u
d
g
Product
d
d
u
(
f
g
)
=
f
d
g
d
u
+
g
d
f
d
u
\frac{d}{du} (fg) = f\frac{dg}{du} + g\frac{df}{du}
d
u
d
(
f
g
)
=
f
d
u
d
g
+
g
d
u
df
Quotient
d
d
u
(
f
g
)
=
g
d
f
d
u
−
f
d
g
d
u
g
2
\frac{d}{du} (\frac{f}{g}) = \frac{g\frac{df}{du} - f\frac{dg}{du}}{g^2}
d
u
d
(
g
f
)
=
g
2
g
d
u
df
−
f
d
u
d
g
Basic formulas
Name of rule
Formula
Constant
d
d
u
(
c
)
=
0
\frac{d}{du} (c) = 0
d
u
d
(
c
)
=
0
Exponential
d
d
u
(
e
u
)
=
e
u
\frac{d}{du} (e^u) = e^u
d
u
d
(
e
u
)
=
e
u
Power
d
d
u
(
u
n
)
=
n
u
n
−
1
\frac{d}{du} (u^n) = nu^{n-1}
d
u
d
(
u
n
)
=
n
u
n
−
1
Logarithmic
d
d
u
(
ln
∣
u
∣
)
=
1
u
\frac{d}{du} (\ln \lvert u \rvert) = \frac{1}{u}
d
u
d
(
ln
∣
u
∣)
=
u
1
Trigonometric