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Product Rule
Related Topics
Wize University Calculus 1 Textbook > Derivatives
The Product Rule
4 Activities
Q:
\textbf{Q:}
Q:
Differentiate
f
(
x
)
=
(
x
3
/
2
)
(
2
x
4
−
5
x
2
−
x
−
1
)
(
x
e
−
π
3.5
)
\displaystyle f(x)=(x^{3/2})\left(2x^{4}-5x^{2}-x^{-1}\right)(x^{e}-\pi^{3.5})
f
(
x
)
=
(
x
3/2
)
(
2
x
4
−
5
x
2
−
x
−
1
)
(
x
e
−
π
3.5
)
2
x
11
2
+
e
−
5
x
7
2
+
e
−
x
1
2
+
e
2x^{\frac{11}{2}+e}-5x^{\frac{7}{2}+e}-x^{\frac{1}{2}+e}
2
x
2
11
+
e
−
5
x
2
7
+
e
−
x
2
1
+
e
3
2
x
1
2
(
8
x
3
−
10
x
+
1
x
2
)
(
e
x
e
−
1
)
\frac{3}{2}x^{\frac{1}{2}}\left(8x^3-10x+\frac{1}{x^2}\right)\left(ex^{e-1}\right)
2
3
x
2
1
(
8
x
3
−
10
x
+
x
2
1
)
(
e
x
e
−
1
)
(
11
x
9
2
−
35
2
x
5
2
−
1
2
x
−
1
2
)
(
x
e
−
π
3.5
)
+
(
2
x
11
2
−
5
7
2
−
x
1
2
)
(
e
x
e
−
1
)
\displaystyle\left(11x^{\frac{9}{2}}-\frac{35}{2}x^{\frac{5}{2}}-\frac{1}{2}x^{-\frac{1}{2}}\right)\left(x^e-\pi^{3.5}\right)+\left(2x^{\frac{11}{2}}-5^{\frac{7}{2}}-x^{\frac{1}{2}}\right)\left(ex^{e-1}\right)
(
11
x
2
9
−
2
35
x
2
5
−
2
1
x
−
2
1
)
(
x
e
−
π
3.5
)
+
(
2
x
2
11
−
5
2
7
−
x
2
1
)
(
e
x
e
−
1
)
I don't know
Check Submission
More The Product Rule Questions:
Product Rule
Suppose
g
(
x
)
g\left(x\right)
g
(
x
)
is differentiable,
f
(
x
)
=
x
2
g
(
x
)
f\left(x\right)=x^2g\left(x\right)
f
(
x
)
=
x
2
g
(
x
)
. Given that
f
′
(
3
)
=
30
f'\left(3\right)=30
f
′
(
3
)
=
30
,
f
′
′
(
3
)
=
19
f''\left(3\right)=19
f
′′
(
3
)
=
19
,
g
(
3
)
=
2
g\left(3\right)=2
g
(
3
)
=
2
, what is
g
′
(
3
)
g'\left(3\right)
g
′
(
3
)
and
g
′
′
(
3
)
g''\left(3\right)
g
′′
(
3
)
?
Product Rule
Suppose
g
(
x
)
g\left(x\right)
g
(
x
)
is differentiable,
f
(
x
)
=
x
2
g
(
x
)
f\left(x\right)=x^2g\left(x\right)
f
(
x
)
=
x
2
g
(
x
)
. Given that
f
′
(
3
)
=
30
f'\left(3\right)=30
f
′
(
3
)
=
30
,
f
′
′
(
3
)
=
19
f''\left(3\right)=19
f
′′
(
3
)
=
19
,
g
(
3
)
=
2
g\left(3\right)=2
g
(
3
)
=
2
, what is
g
′
(
3
)
g'\left(3\right)
g
′
(
3
)
and
g
′
′
(
3
)
g''\left(3\right)
g
′′
(
3
)
?
Product Rule
Suppose
g
(
x
)
g\left(x\right)
g
(
x
)
is differentiable,
f
(
x
)
=
x
2
g
(
x
)
f\left(x\right)=x^2g\left(x\right)
f
(
x
)
=
x
2
g
(
x
)
. Given that
f
′
(
3
)
=
30
f'\left(3\right)=30
f
′
(
3
)
=
30
,
f
′
′
(
3
)
=
19
f''\left(3\right)=19
f
′′
(
3
)
=
19
,
g
(
3
)
=
2
g\left(3\right)=2
g
(
3
)
=
2
, what is
g
′
(
3
)
g'\left(3\right)
g
′
(
3
)
and
g
′
′
(
3
)
g''\left(3\right)
g
′′
(
3
)
?
Product Rule
Suppose
g
(
x
)
g\left(x\right)
g
(
x
)
is differentiable,
f
(
x
)
=
x
2
g
(
x
)
f\left(x\right)=x^2g\left(x\right)
f
(
x
)
=
x
2
g
(
x
)
. Given that
f
′
(
3
)
=
30
f'\left(3\right)=30
f
′
(
3
)
=
30
,
f
′
′
(
3
)
=
19
f''\left(3\right)=19
f
′′
(
3
)
=
19
,
g
(
3
)
=
2
g\left(3\right)=2
g
(
3
)
=
2
, what is
g
′
(
3
)
g'\left(3\right)
g
′
(
3
)
and
g
′
′
(
3
)
g''\left(3\right)
g
′′
(
3
)
?
Product Rule
Suppose
g
(
x
)
g\left(x\right)
g
(
x
)
is differentiable,
f
(
x
)
=
x
2
g
(
x
)
f\left(x\right)=x^2g\left(x\right)
f
(
x
)
=
x
2
g
(
x
)
. Given that
f
′
(
3
)
=
30
f'\left(3\right)=30
f
′
(
3
)
=
30
,
f
′
′
(
3
)
=
19
f''\left(3\right)=19
f
′′
(
3
)
=
19
,
g
(
3
)
=
2
g\left(3\right)=2
g
(
3
)
=
2
, what is
g
′
(
3
)
g'\left(3\right)
g
′
(
3
)
and
g
′
′
(
3
)
g''\left(3\right)
g
′′
(
3
)
?
Product Rule
Suppose
g
(
x
)
g\left(x\right)
g
(
x
)
is differentiable,
f
(
x
)
=
x
2
g
(
x
)
f\left(x\right)=x^2g\left(x\right)
f
(
x
)
=
x
2
g
(
x
)
. Given that
f
′
(
3
)
=
30
f'\left(3\right)=30
f
′
(
3
)
=
30
,
f
′
′
(
3
)
=
19
f''\left(3\right)=19
f
′′
(
3
)
=
19
,
g
(
3
)
=
2
g\left(3\right)=2
g
(
3
)
=
2
, what is
g
′
(
3
)
g'\left(3\right)
g
′
(
3
)
and
g
′
′
(
3
)
g''\left(3\right)
g
′′
(
3
)
?
Product Rule
Suppose
g
(
x
)
g\left(x\right)
g
(
x
)
is differentiable,
f
(
x
)
=
x
2
g
(
x
)
f\left(x\right)=x^2g\left(x\right)
f
(
x
)
=
x
2
g
(
x
)
. Given that
f
′
(
3
)
=
30
f'\left(3\right)=30
f
′
(
3
)
=
30
,
f
′
′
(
3
)
=
19
f''\left(3\right)=19
f
′′
(
3
)
=
19
,
g
(
3
)
=
2
g\left(3\right)=2
g
(
3
)
=
2
, what is
g
′
(
3
)
g'\left(3\right)
g
′
(
3
)
and
g
′
′
(
3
)
g''\left(3\right)
g
′′
(
3
)
?
Product Rule
Suppose
g
(
x
)
g\left(x\right)
g
(
x
)
is differentiable,
f
(
x
)
=
x
2
g
(
x
)
f\left(x\right)=x^2g\left(x\right)
f
(
x
)
=
x
2
g
(
x
)
. Given that
f
′
(
3
)
=
30
f'\left(3\right)=30
f
′
(
3
)
=
30
,
f
′
′
(
3
)
=
19
f''\left(3\right)=19
f
′′
(
3
)
=
19
,
g
(
3
)
=
2
g\left(3\right)=2
g
(
3
)
=
2
, what is
g
′
(
3
)
g'\left(3\right)
g
′
(
3
)
and
g
′
′
(
3
)
g''\left(3\right)
g
′′
(
3
)
?
Product Rule
Suppose
g
(
x
)
g\left(x\right)
g
(
x
)
is differentiable,
f
(
x
)
=
x
2
g
(
x
)
f\left(x\right)=x^2g\left(x\right)
f
(
x
)
=
x
2
g
(
x
)
. Given that
f
′
(
3
)
=
30
f'\left(3\right)=30
f
′
(
3
)
=
30
,
f
′
′
(
3
)
=
19
f''\left(3\right)=19
f
′′
(
3
)
=
19
,
g
(
3
)
=
2
g\left(3\right)=2
g
(
3
)
=
2
, what is
g
′
(
3
)
g'\left(3\right)
g
′
(
3
)
and
g
′
′
(
3
)
g''\left(3\right)
g
′′
(
3
)
?
Product Rule
Suppose
g
(
x
)
g\left(x\right)
g
(
x
)
is differentiable,
f
(
x
)
=
x
2
g
(
x
)
f\left(x\right)=x^2g\left(x\right)
f
(
x
)
=
x
2
g
(
x
)
. Given that
f
′
(
3
)
=
30
f'\left(3\right)=30
f
′
(
3
)
=
30
,
f
′
′
(
3
)
=
19
f''\left(3\right)=19
f
′′
(
3
)
=
19
,
g
(
3
)
=
2
g\left(3\right)=2
g
(
3
)
=
2
, what is
g
′
(
3
)
g'\left(3\right)
g
′
(
3
)
and
g
′
′
(
3
)
g''\left(3\right)
g
′′
(
3
)
?
Product Rule
Suppose
g
(
x
)
g\left(x\right)
g
(
x
)
is differentiable,
f
(
x
)
=
x
2
g
(
x
)
f\left(x\right)=x^2g\left(x\right)
f
(
x
)
=
x
2
g
(
x
)
. Given that
f
′
(
3
)
=
30
f'\left(3\right)=30
f
′
(
3
)
=
30
,
f
′
′
(
3
)
=
19
f''\left(3\right)=19
f
′′
(
3
)
=
19
,
g
(
3
)
=
2
g\left(3\right)=2
g
(
3
)
=
2
, what is
g
′
(
3
)
g'\left(3\right)
g
′
(
3
)
and
g
′
′
(
3
)
g''\left(3\right)
g
′′
(
3
)
?
Product Rule
Suppose
g
(
x
)
g\left(x\right)
g
(
x
)
is differentiable,
f
(
x
)
=
x
2
g
(
x
)
f\left(x\right)=x^2g\left(x\right)
f
(
x
)
=
x
2
g
(
x
)
. Given that
f
′
(
3
)
=
30
f'\left(3\right)=30
f
′
(
3
)
=
30
,
f
′
′
(
3
)
=
19
f''\left(3\right)=19
f
′′
(
3
)
=
19
,
g
(
3
)
=
2
g\left(3\right)=2
g
(
3
)
=
2
, what is
g
′
(
3
)
g'\left(3\right)
g
′
(
3
)
and
g
′
′
(
3
)
g''\left(3\right)
g
′′
(
3
)
?
Product Rule
Suppose
g
(
x
)
g\left(x\right)
g
(
x
)
is differentiable,
f
(
x
)
=
x
2
g
(
x
)
f\left(x\right)=x^2g\left(x\right)
f
(
x
)
=
x
2
g
(
x
)
. Given that
f
′
(
3
)
=
30
f'\left(3\right)=30
f
′
(
3
)
=
30
,
f
′
′
(
3
)
=
19
f''\left(3\right)=19
f
′′
(
3
)
=
19
,
g
(
3
)
=
2
g\left(3\right)=2
g
(
3
)
=
2
, what is
g
′
(
3
)
g'\left(3\right)
g
′
(
3
)
and
g
′
′
(
3
)
g''\left(3\right)
g
′′
(
3
)
?
Product Rule
Suppose
g
(
x
)
g\left(x\right)
g
(
x
)
is differentiable,
f
(
x
)
=
x
2
g
(
x
)
f\left(x\right)=x^2g\left(x\right)
f
(
x
)
=
x
2
g
(
x
)
. Given that
f
′
(
3
)
=
30
f'\left(3\right)=30
f
′
(
3
)
=
30
,
f
′
′
(
3
)
=
19
f''\left(3\right)=19
f
′′
(
3
)
=
19
,
g
(
3
)
=
2
g\left(3\right)=2
g
(
3
)
=
2
, what is
g
′
(
3
)
g'\left(3\right)
g
′
(
3
)
and
g
′
′
(
3
)
g''\left(3\right)
g
′′
(
3
)
?
Product Rule
Suppose
g
(
x
)
g\left(x\right)
g
(
x
)
is differentiable,
f
(
x
)
=
x
2
g
(
x
)
f\left(x\right)=x^2g\left(x\right)
f
(
x
)
=
x
2
g
(
x
)
. Given that
f
′
(
3
)
=
30
f'\left(3\right)=30
f
′
(
3
)
=
30
,
f
′
′
(
3
)
=
19
f''\left(3\right)=19
f
′′
(
3
)
=
19
,
g
(
3
)
=
2
g\left(3\right)=2
g
(
3
)
=
2
, what is
g
′
(
3
)
g'\left(3\right)
g
′
(
3
)
and
g
′
′
(
3
)
g''\left(3\right)
g
′′
(
3
)
?
Quotient and Product
Find the derivative of
g
(
t
)
=
(
1
+
2
t
7
t
)
(
t
2
−
1
)
\displaystyle g(t)=\left(\frac{1+2t}{7t}\right)(t^{2}-1)
g
(
t
)
=
(
7
t
1
+
2
t
)
(
t
2
−
1
)
The product rule
Find the derivative of
f
(
x
)
=
x
3
e
x
f(x) = x^3 e^x
f
(
x
)
=
x
3
e
x
The graph of w(x) is given below.
Practice: Product Rule*
Suppose
g
(
x
)
g\left(x\right)
g
(
x
)
is differentiable,
f
(
x
)
=
x
2
g
(
x
)
f\left(x\right)=x^2g\left(x\right)
f
(
x
)
=
x
2
g
(
x
)
. Given that
f
′
(
3
)
=
30
f'\left(3\right)=30
f
′
(
3
)
=
30
,
f
′
′
(
3
)
=
19
f''\left(3\right)=19
f
′′
(
3
)
=
19
,
g
(
3
)
=
2
g\left(3\right)=2
g
(
3
)
=
2
, what is
g
′
(
3
)
g'\left(3\right)
g
′
(
3
)
and
g
′
′
(
3
)
g''\left(3\right)
g
′′
(
3
)
?
Product Rule
Differentiate
f
(
x
)
=
(
x
3
/
2
)
(
2
x
4
−
5
x
2
−
x
−
1
)
(
x
e
−
π
3.5
)
\displaystyle f(x)=(x^{3/2})\left(2x^{4}-5x^{2}-x^{-1}\right)(x^{e}-\pi^{3.5})
f
(
x
)
=
(
x
3/2
)
(
2
x
4
−
5
x
2
−
x
−
1
)
(
x
e
−
π
3.5
)
Quotient and Product
Find the derivative of
g
(
t
)
=
(
1
+
2
t
7
t
)
(
t
2
−
1
)
\displaystyle g(t)=\left(\frac{1+2t}{7t}\right)(t^{2}-1)
g
(
t
)
=
(
7
t
1
+
2
t
)
(
t
2
−
1
)
Find the derivative of
f
(
x
)
=
(
x
3
/
2
+
2
x
)
(
x
7
/
4
+
x
)
\displaystyle f(x)=\left(x^{3/2}+\frac{2}{x}\right)\left(x^{7/4}+x\right)
f
(
x
)
=
(
x
3/2
+
x
2
)
(
x
7/4
+
x
)
.
Product Rule
Suppose
g
(
x
)
g\left(x\right)
g
(
x
)
is differentiable,
f
(
x
)
=
x
2
g
(
x
)
f\left(x\right)=x^2g\left(x\right)
f
(
x
)
=
x
2
g
(
x
)
. Given that
f
′
(
3
)
=
30
f'\left(3\right)=30
f
′
(
3
)
=
30
,
f
′
′
(
3
)
=
19
f''\left(3\right)=19
f
′′
(
3
)
=
19
,
g
(
3
)
=
2
g\left(3\right)=2
g
(
3
)
=
2
, what is
g
′
(
3
)
g'\left(3\right)
g
′
(
3
)
and
g
′
′
(
3
)
g''\left(3\right)
g
′′
(
3
)
?