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Find the derivative of f(x)=(x^3/2+2/x)(x^7/4+x) .
Related Topics
Wize University Calculus 1 Textbook > Derivatives
The Product Rule
4 Activities
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
)
.
13
4
x
9
/
4
+
5
2
x
3
/
2
+
3
2
x
−
1
/
4
\dfrac{13}{4}x^{9/4}+\dfrac{5}{2}x^{3/2}+\dfrac{3}{2}x^{-1/4}
4
13
x
9/4
+
2
5
x
3/2
+
2
3
x
−
1/4
13
4
x
9
/
4
+
5
2
x
3
/
2
+
3
2
x
1
/
4
\dfrac{13}{4}x^{9/4}+\dfrac{5}{2}x^{3/2}+\dfrac{3}{2}x^{1/4}
4
13
x
9/4
+
2
5
x
3/2
+
2
3
x
1/4
13
4
x
9
/
4
+
5
2
x
−
3
/
2
+
3
2
x
1
/
4
\dfrac{13}{4}x^{9/4}+\dfrac{5}{2}x^{-3/2}+\dfrac{3}{2}x^{1/4}
4
13
x
9/4
+
2
5
x
−
3/2
+
2
3
x
1/4
13
4
x
−
9
/
4
+
5
2
x
3
/
2
+
3
2
x
−
1
/
4
\dfrac{13}{4}x^{-9/4}+\dfrac{5}{2}x^{3/2}+\dfrac{3}{2}x^{-1/4}
4
13
x
−
9/4
+
2
5
x
3/2
+
2
3
x
−
1/4
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
)
?
Product Rule
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
)
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
)
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
)
?