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Compute the derivative of f(x) =(x^2 + 3x^4/3 + 1)/(x^2 + 1).
Related Topics
Wize University Calculus 1 Textbook > Derivatives
The Quotient Rule
3 Activities
Compute the derivative of
f
(
x
)
=
x
2
+
3
x
4
/
3
+
1
x
2
+
1
f(x) =\dfrac{x^2 + 3x^{4/3} + 1}{x^2 + 1}
f
(
x
)
=
x
2
+
1
x
2
+
3
x
4/3
+
1
.
2
x
1
/
3
(
x
2
−
2
)
(
x
2
+
1
)
2
\dfrac{2x^{1/3}(x^2 - 2)}{(x^2 + 1)^2}
(
x
2
+
1
)
2
2
x
1/3
(
x
2
−
2
)
−
2
x
4
/
3
(
x
2
−
1
)
(
x
2
+
1
)
2
-\dfrac{2x^{4/3}(x^2 - 1)}{(x^2 + 1)^2}
−
(
x
2
+
1
)
2
2
x
4/3
(
x
2
−
1
)
−
2
x
1
/
3
(
x
2
−
2
)
(
x
2
+
1
)
2
-\dfrac{2x^{1/3}(x^2 - 2)}{(x^2 + 1)^2}
−
(
x
2
+
1
)
2
2
x
1/3
(
x
2
−
2
)
2
x
4
/
3
(
x
2
−
1
)
(
x
2
+
1
)
2
\dfrac{2x^{4/3}(x^2 - 1)}{(x^2 + 1)^2}
(
x
2
+
1
)
2
2
x
4/3
(
x
2
−
1
)
None of the above
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Check Submission
More The Quotient Rule Questions:
Derivatives
Find the derivative of the following functions at the given values
a)
f
(
x
)
=
arcsin
(
x
2
)
x
3
−
1
f\left(x\right)=\frac{\arcsin\left(x^2\right)}{x^3-1}
f
(
x
)
=
x
3
−
1
a
r
c
s
i
n
(
x
2
)
at
x
=
0
x=0
x
=
0
b)
g
(
x
)
=
(
cos
x
)
e
x
g\left(x\right)=\left(\cos x\right)^{e^x}
g
(
x
)
=
(
cos
x
)
e
x
at
x
=
0
x=0
x
=
0
The quotient rule
Is
u
′
(
t
)
=
2
(
t
−
1
)
3
(
−
3
t
+
23
)
(
3
t
+
7
)
7
\displaystyle u'\left(t\right)=\frac{2\left(t-1\right)^3\left(-3t+23\right)}{\left(3t+7\right)^7}
u
′
(
t
)
=
(
3
t
+
7
)
7
2
(
t
−
1
)
3
(
−
3
t
+
23
)
the derivative of
u
(
t
)
=
(
t
−
1
)
4
(
3
t
+
7
)
6
\displaystyle u\left(t\right)=\frac{\left(t-1\right)^4}{\left(3t+7\right)^6}
u
(
t
)
=
(
3
t
+
7
)
6
(
t
−
1
)
4
?
The quotient rule
Find the derivative of the following function:
g
(
x
)
=
x
+
1
x
+
1
\displaystyle g(x)=\frac{\sqrt{x+1}}{\sqrt{x}+1}
g
(
x
)
=
x
+
1
x
+
1
The quotient rule
Is
u
′
(
t
)
=
2
(
t
−
1
)
3
(
−
3
t
+
23
)
(
3
t
+
7
)
7
\displaystyle u'\left(t\right)=\frac{2\left(t-1\right)^3\left(-3t+23\right)}{\left(3t+7\right)^7}
u
′
(
t
)
=
(
3
t
+
7
)
7
2
(
t
−
1
)
3
(
−
3
t
+
23
)
the derivative of
u
(
t
)
=
(
t
−
1
)
4
(
3
t
+
7
)
6
\displaystyle u\left(t\right)=\frac{\left(t-1\right)^4}{\left(3t+7\right)^6}
u
(
t
)
=
(
3
t
+
7
)
6
(
t
−
1
)
4
?
The quotient rule
Given the function
f
(
x
)
=
g
(
x
)
h
(
x
)
−
e
x
cos
x
f\left(x\right)=\dfrac{g\left(x\right)}{h\left(x\right)}-e^x\cos x
f
(
x
)
=
h
(
x
)
g
(
x
)
−
e
x
cos
x
, and given
g
(
π
2
)
=
2
g\left(\dfrac{\pi}{2}\right)=2
g
(
2
π
)
=
2
,
g
′
(
π
2
)
=
3
g'\left(\dfrac{\pi}{2}\right)=3
g
′
(
2
π
)
=
3
,
h
(
π
2
)
=
1
h\left(\dfrac{\pi}{2}\right)=1
h
(
2
π
)
=
1
, and
h
′
(
π
2
)
=
2
h'\left(\dfrac{\pi}{2}\right)=2
h
′
(
2
π
)
=
2
, find
f
′
(
π
2
)
f'\left(\dfrac{\pi}{2}\right)
f
′
(
2
π
)
.
Derivatives: Exponential and Logarithmic Functions
Find
f
′
(
x
)
f'(x)
f
′
(
x
)
if
f
(
x
)
=
ln
x
x
e
x
\displaystyle f\left(x\right)=\frac{\ln x}{xe^{x}}
f
(
x
)
=
x
e
x
ln
x
. Simplify.
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
)
Practice: Quotient Rule
Q.
\textbf{Q.}
Q.
Find the derivative of
f
(
x
)
=
x
2
+
3
x
4
/
3
+
1
x
2
+
1
\displaystyle f(x)=\frac{x^2+3x^{4/3}+1}{x^2+1}
f
(
x
)
=
x
2
+
1
x
2
+
3
x
4/3
+
1
Find the derivative of the following function:
g
(
x
)
=
x
+
1
x
+
1
\displaystyle g(x)=\frac{\sqrt{x+1}}{\sqrt{x}+1}
g
(
x
)
=
x
+
1
x
+
1
The Quotient and Chain Rules
Let
f
(
x
)
=
x
2
−
6
x
−
3
\displaystyle f(x) = \frac{\sqrt{x^2 - 6}}{x - 3}
f
(
x
)
=
x
−
3
x
2
−
6
.
The Quotient Rule
Find the derivative of
g
(
x
)
=
x
2
−
5
2
x
+
1
g(x) = \frac{x^2 - 5}{2x + 1}
g
(
x
)
=
2
x
+
1
x
2
−
5
Derivatives
Find the derivative of the following functions at the given values
a)
f
(
x
)
=
arcsin
(
x
2
)
x
3
−
1
f\left(x\right)=\frac{\arcsin\left(x^2\right)}{x^3-1}
f
(
x
)
=
x
3
−
1
a
r
c
s
i
n
(
x
2
)
at
x
=
0
x=0
x
=
0
b)
g
(
x
)
=
(
cos
x
)
e
x
g\left(x\right)=\left(\cos x\right)^{e^x}
g
(
x
)
=
(
cos
x
)
e
x
at
x
=
0
x=0
x
=
0
The Quotient Rule
Compute the derivative of
f
(
x
)
=
x
2
+
3
x
4
/
3
+
1
x
2
+
1
f(x)=\frac{x^2+3x^{4/3}+1}{x^2+1}
f
(
x
)
=
x
2
+
1
x
2
+
3
x
4/3
+
1
Practice: Product Rule*
Suppose
g
(
x
)
g\left(x\right)
g
(
x
)
is differentiable,
f
(
x
)
=
1
+
x
g
(
x
)
x
2
f\left(x\right)=\frac{1+x\ g\left(x\right)}{x^2}
f
(
x
)
=
x
2
1
+
x
g
(
x
)
,
f
′
(
1
)
=
2
f'\left(1\right)=2
f
′
(
1
)
=
2
,
g
(
1
)
=
1
g\left(1\right)=1
g
(
1
)
=
1
, what is
g
′
(
1
)
g'\left(1\right)
g
′
(
1
)
?
The quotient rule
Given the function
f
(
x
)
=
g
(
x
)
h
(
x
)
−
e
x
cos
x
f\left(x\right)=\dfrac{g\left(x\right)}{h\left(x\right)}-e^x\cos x
f
(
x
)
=
h
(
x
)
g
(
x
)
−
e
x
cos
x
, and given
g
(
π
2
)
=
2
g\left(\dfrac{\pi}{2}\right)=2
g
(
2
π
)
=
2
,
g
′
(
π
2
)
=
3
g'\left(\dfrac{\pi}{2}\right)=3
g
′
(
2
π
)
=
3
,
h
(
π
2
)
=
1
h\left(\dfrac{\pi}{2}\right)=1
h
(
2
π
)
=
1
, and
h
′
(
π
2
)
=
2
h'\left(\dfrac{\pi}{2}\right)=2
h
′
(
2
π
)
=
2
, find
f
′
(
π
2
)
f'\left(\dfrac{\pi}{2}\right)
f
′
(
2
π
)
.
Derivatives: Exponential and Logarithmic Functions
Find
f
′
(
x
)
f'(x)
f
′
(
x
)
if
f
(
x
)
=
ln
x
x
e
x
\displaystyle f\left(x\right)=\frac{\ln x}{xe^{x}}
f
(
x
)
=
x
e
x
ln
x
. Simplify.
The quotient rule
Is
u
′
(
t
)
=
2
(
t
−
1
)
3
(
−
3
t
+
23
)
(
3
t
+
7
)
7
\displaystyle u'\left(t\right)=\frac{2\left(t-1\right)^3\left(-3t+23\right)}{\left(3t+7\right)^7}
u
′
(
t
)
=
(
3
t
+
7
)
7
2
(
t
−
1
)
3
(
−
3
t
+
23
)
the derivative of
u
(
t
)
=
(
t
−
1
)
4
(
3
t
+
7
)
6
\displaystyle u\left(t\right)=\frac{\left(t-1\right)^4}{\left(3t+7\right)^6}
u
(
t
)
=
(
3
t
+
7
)
6
(
t
−
1
)
4
?
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
)
Practice: Quotient Rule
Find the derivative of
f
(
x
)
=
x
2
+
3
x
4
/
3
+
1
x
2
+
1
\displaystyle f(x)=\frac{x^2+3x^{4/3}+1}{x^2+1}
f
(
x
)
=
x
2
+
1
x
2
+
3
x
4/3
+
1
The Quotient Rule
If
h
(
x
)
=
f
(
x
)
−
1
x
+
2
h\left(x\right)=\frac{f\left(x\right)-1}{x+2}
h
(
x
)
=
x
+
2
f
(
x
)
−
1
,
f
(
3
)
=
2
and
f
′
(
3
)
=
−
6
f\left(3\right)=2\ \text{and}\ f'\left(3\right)=-6
f
(
3
)
=
2
and
f
′
(
3
)
=
−
6
, calculate the value of
h
′
(
3
)
h'\left(3\right)
h
′
(
3
)
The Quotient Rule
Is
u
′
(
t
)
=
2
(
t
−
1
)
3
(
−
3
t
+
23
)
(
3
t
+
7
)
7
u'\left(t\right)=\frac{2\left(t-1\right)^3\left(-3t+23\right)}{\left(3t+7\right)^7}
u
′
(
t
)
=
(
3
t
+
7
)
7
2
(
t
−
1
)
3
(
−
3
t
+
23
)
the derivative of
u
(
t
)
=
(
t
−
1
)
4
(
3
t
+
7
)
6
u\left(t\right)=\frac{\left(t-1\right)^4}{\left(3t+7\right)^6}
u
(
t
)
=
(
3
t
+
7
)
6
(
t
−
1
)
4
?
Find the derivative with respect to
x
x
x
of the function
x
2030
3
+
x
2030
\displaystyle \frac{x^{2030}}{3+x^{2030}}
3
+
x
2030
x
2030
.
The Quotient Rule
Suppose
g
(
x
)
g\left(x\right)
g
(
x
)
is differentiable,
f
(
x
)
=
1
+
x
g
(
x
)
x
2
f\left(x\right)=\frac{1+x\ g\left(x\right)}{x^2}
f
(
x
)
=
x
2
1
+
x
g
(
x
)
,
f
′
(
1
)
=
2
f'\left(1\right)=2
f
′
(
1
)
=
2
,
g
(
1
)
=
1
g\left(1\right)=1
g
(
1
)
=
1
, what is
g
′
(
1
)
g'\left(1\right)
g
′
(
1
)
?
Find the derivative of the following function:
g
(
x
)
=
x
+
1
x
+
1
\displaystyle g(x)=\frac{\sqrt{x+1}}{\sqrt{x}+1}
g
(
x
)
=
x
+
1
x
+
1
The quotient rule
Find the derivative of the following function:
g
(
x
)
=
x
+
1
x
+
1
\displaystyle g(x)=\frac{\sqrt{x+1}}{\sqrt{x}+1}
g
(
x
)
=
x
+
1
x
+
1