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Derivatives
Derivatives
10 formulas · Mathematics
Definition of the derivative
f
′
(
x
0
)
=
lim
h
→
0
f
(
x
0
+
h
)
−
f
(
x
0
)
h
f'(x_0)=\lim_{h\to 0}\dfrac{f(x_0+h)-f(x_0)}{h}
f
′
(
x
0
)
=
h
→
0
lim
h
f
(
x
0
+
h
)
−
f
(
x
0
)
Derivative of a power
(
x
n
)
′
=
n
x
n
−
1
(x^n)'=n x^{n-1}
(
x
n
)
′
=
n
x
n
−
1
Derivative of a constant
(
c
)
′
=
0
(c)'=0
(
c
)
′
=
0
Derivative of a sum
(
f
±
g
)
′
=
f
′
±
g
′
(f\pm g)'=f'\pm g'
(
f
±
g
)
′
=
f
′
±
g
′
Derivative of a product
(
f
g
)
′
=
f
′
g
+
f
g
′
(fg)'=f'g+fg'
(
f
g
)
′
=
f
′
g
+
f
g
′
Derivative of a quotient
(
f
g
)
′
=
f
′
g
−
f
g
′
g
2
\left(\tfrac{f}{g}\right)'=\dfrac{f'g-fg'}{g^2}
(
g
f
)
′
=
g
2
f
′
g
−
f
g
′
Equation of the tangent line to a graph
y
=
f
′
(
x
0
)
(
x
−
x
0
)
+
f
(
x
0
)
y=f'(x_0)(x-x_0)+f(x_0)
y
=
f
′
(
x
0
)
(
x
−
x
0
)
+
f
(
x
0
)
Definite integral (area under a curve)
P
=
∫
a
b
f
(
x
)
d
x
P=\int_a^b f(x)\,dx
P
=
∫
a
b
f
(
x
)
d
x
Derivatives of elementary functions
(
sin
x
)
′
=
cos
x
,
(
cos
x
)
′
=
−
sin
x
,
(
e
x
)
′
=
e
x
,
(
ln
x
)
′
=
1
x
(\sin x)'=\cos x,\ (\cos x)'=-\sin x,\ (e^x)'=e^x,\ (\ln x)'=\tfrac1x
(
sin
x
)
′
=
cos
x
,
(
cos
x
)
′
=
−
sin
x
,
(
e
x
)
′
=
e
x
,
(
ln
x
)
′
=
x
1
Basic integrals
∫
x
n
d
x
=
x
n
+
1
n
+
1
+
C
,
∫
1
x
d
x
=
ln
∣
x
∣
+
C
\int x^n\,dx=\tfrac{x^{n+1}}{n+1}+C,\ \int\tfrac1x\,dx=\ln|x|+C
∫
x
n
d
x
=
n
+
1
x
n
+
1
+
C
,
∫
x
1
d
x
=
ln
∣
x
∣
+
C
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