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Product Rule
Reference
> Calculus: Differentiation
Description
(
f
g
)
′
=
f
′
g
+
f
g
′
(fg)'=f'g+fg'
(
f
g
)
′
=
f
′
g
+
f
g
′
Examples
d
d
x
sin
x
x
2
\frac{d}{dx} \sin{x}{x}^{2}
d
x
d
sin
x
x
2
1
Regroup terms.
d
d
x
x
2
sin
x
\frac{d}{dx} {x}^{2}\sin{x}
d
x
d
x
2
sin
x
2
Use
Product Rule
to find the derivative of
x
2
sin
x
{x}^{2}\sin{x}
x
2
sin
x
. The product rule states that
(
f
g
)
′
=
f
′
g
+
f
g
′
(fg)'=f'g+fg'
(
f
g
)
′
=
f
′
g
+
f
g
′
.
(
d
d
x
x
2
)
sin
x
+
x
2
(
d
d
x
sin
x
)
(\frac{d}{dx} {x}^{2})\sin{x}+{x}^{2}(\frac{d}{dx} \sin{x})
(
d
x
d
x
2
)
sin
x
+
x
2
(
d
x
d
sin
x
)
3
Use
Power Rule
:
d
d
x
x
n
=
n
x
n
−
1
\frac{d}{dx} {x}^{n}=n{x}^{n-1}
d
x
d
x
n
=
n
x
n
−
1
.
2
x
sin
x
+
x
2
(
d
d
x
sin
x
)
2x\sin{x}+{x}^{2}(\frac{d}{dx} \sin{x})
2
x
sin
x
+
x
2
(
d
x
d
sin
x
)
4
Use
Trigonometric Differentiation
: the derivative of
sin
x
\sin{x}
sin
x
is
cos
x
\cos{x}
cos
x
.
2
x
sin
x
+
x
2
cos
x
2x\sin{x}+{x}^{2}\cos{x}
2
x
sin
x
+
x
2
cos
x
Done
2*x*sin(x)+x^2*cos(x)
See Also
-
Quotient Rule
-
Sum Rule
English
Product Rule