WOLFRAM|DEMONSTRATIONS PROJECT

The Product Rule

​
f(x)
ln(x)
g(x)
cos(x)
D
x
​(
f(x)
g(x)
) =
g(x)
′
f
(x)
+
f(x)
′
g
(x)
=
cos(x)
1
x
+
log(x)
-sin(x)
=
cos(x)
x
-log(x)sin(x)
If
f
and
g
are both differentiable at
x
, then the derivative of their product at
x
is given by the product rule shown above. On the graph,
f
is blue,
g
is red,
f(x)g(x)
is purple, and the derivative of
f(x)g(x)
is thick purple.