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Derivatives: Logarithmic Functions
Related Topics
Wize University Calculus 1 Textbook > Derivatives
Derivatives of Logarithmic Functions
5 Activities
Calculate the derivative of the following functions.
f
(
x
)
=
ln
x
x
+
x
e
2
x
\displaystyle f(x)=\frac{\ln{x}}{x+xe^{2x}}
f
(
x
)
=
x
+
x
e
2
x
ln
x
1
/
x
(
x
+
x
e
2
x
)
−
ln
(
x
)
(
1
+
e
2
x
+
2
x
e
2
x
)
(
x
−
x
e
2
x
)
2
\dfrac{1/x(x+xe^{2x})-\ln(x)(1+e^{2x}+2xe^{2x})}{(x-xe^{2x})^2}
(
x
−
x
e
2
x
)
2
1/
x
(
x
+
x
e
2
x
)
−
ln
(
x
)
(
1
+
e
2
x
+
2
x
e
2
x
)
1
/
x
(
x
+
x
e
2
x
)
+
ln
(
x
)
(
1
+
e
2
x
+
2
x
e
2
x
)
(
x
+
x
e
2
x
)
2
\dfrac{1/x(x+xe^{2x})+\ln(x)(1+e^{2x}+2xe^{2x})}{(x+xe^{2x})^2}
(
x
+
x
e
2
x
)
2
1/
x
(
x
+
x
e
2
x
)
+
ln
(
x
)
(
1
+
e
2
x
+
2
x
e
2
x
)
1
/
x
(
x
+
x
e
2
x
)
−
ln
(
x
)
(
1
+
e
2
x
+
2
x
e
2
x
)
(
x
+
x
e
2
x
)
2
\dfrac{1/x(x+xe^{2x})-\ln(x)(1+e^{2x}+2xe^{2x})}{(x+xe^{2x})^2}
(
x
+
x
e
2
x
)
2
1/
x
(
x
+
x
e
2
x
)
−
ln
(
x
)
(
1
+
e
2
x
+
2
x
e
2
x
)
1
/
x
(
x
+
x
e
2
x
)
−
ln
(
x
)
(
1
+
e
2
x
−
2
x
e
2
x
)
(
x
+
x
e
2
x
)
2
\dfrac{1/x(x+xe^{2x})-\ln(x)(1+e^{2x}-2xe^{2x})}{(x+xe^{2x})^2}
(
x
+
x
e
2
x
)
2
1/
x
(
x
+
x
e
2
x
)
−
ln
(
x
)
(
1
+
e
2
x
−
2
x
e
2
x
)
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Check Submission
More Derivatives of Logarithmic Functions Questions:
Derivatives: Logarithmic Functions
Find the derivative of
f
(
x
)
=
e
x
x
2
+
ln
(
x
)
3
f(x) = e^{x} x^2 + \frac{\ln(x)}{3}
f
(
x
)
=
e
x
x
2
+
3
ln
(
x
)
Practice: Chain Rule
Find the derivative of
y
=
ln
(
arctan
x
)
y=\ln(\arctan x)
y
=
ln
(
arctan
x
)
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.
Inverse Trig with Log
Compute the derivative of
f
(
x
)
=
arctan
(
log
10
x
)
\displaystyle f(x)=\text{arctan}\left(\text{log}_{10}x\right)
f
(
x
)
=
arctan
(
log
10
x
)
Applying Rules of Logs
Find
d
y
d
x
\displaystyle\frac{dy}{dx}
d
x
d
y
given
y
=
log
6
(
x
−
1
)
5
(
x
+
1
)
10
3
\displaystyle y=\log_6 \sqrt[3]{\frac{(x-1)^{5}}{(x+1)^{10}}}
y
=
lo
g
6
3
(
x
+
1
)
10
(
x
−
1
)
5
Practice: Log and Ln
Q.
\textbf{Q.}
Q.
Find the derivative of
y
=
log
3
(
7
x
ln
2
)
\displaystyle y=\text{log}_3(7x^{\ln2})
y
=
log
3
(
7
x
l
n
2
)
Derivatives of Logarithmic Functions: Two Logs
Find
d
y
/
d
x
dy/dx
d
y
/
d
x
for
y
=
5
log
5
(
log
2
t
)
\displaystyle y=5\log_{5}(\log_{2}t)
y
=
5
lo
g
5
(
lo
g
2
t
)
Derivatives: Logarithmic Functions
Compute the derivative of
f
(
x
)
=
x
x
+
1
f(x) = x^{x + 1}
f
(
x
)
=
x
x
+
1
. Remember that
log
x
=
log
e
x
=
ln
x
\log x = \log_e x = \ln x
lo
g
x
=
lo
g
e
x
=
ln
x
.
The chain rule: Logarithmic Derivatives
The derivative of
log
3
(
e
3
x
)
\log_3\left(e^{3x}\right)
lo
g
3
(
e
3
x
)
is
Calculate the derivative of the following
f
(
x
)
=
ln
x
x
+
x
e
2
x
\displaystyle f(x)=\frac{\ln{x}}{x+xe^{2x}}
f
(
x
)
=
x
+
x
e
2
x
ln
x
Derivatives: Logarithmic Functions
Find the derivative of the following function.
f
(
x
)
=
ln
(
x
2
+
x
)
\displaystyle f(x)=\ln{(x^2+\sqrt{x})}
f
(
x
)
=
ln
(
x
2
+
x
)
Derivatives: Logarithmic Functions
Compute the derivative of
f
(
x
)
=
log
2
(
x
2
+
2
x
e
x
)
f(x)=\log_2(x^2+2xe^x)
f
(
x
)
=
lo
g
2
(
x
2
+
2
x
e
x
)
Differentiate the following:
y
=
e
x
2
+
ln
x
x
\begin{aligned} &y=\frac{e^{x^2+\ln x}}{x} \end{aligned}
y
=
x
e
x
2
+
l
n
x
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.
Inverse Trig with Log
Compute the derivative of
f
(
x
)
=
arctan
(
log
10
x
)
\displaystyle f(x)=\text{arctan}\left(\text{log}_{10}x\right)
f
(
x
)
=
arctan
(
log
10
x
)
Applying Rules of Logs
Find
d
y
d
x
\displaystyle\frac{dy}{dx}
d
x
d
y
given
y
=
log
6
(
x
−
1
)
5
(
x
+
1
)
10
3
\displaystyle y=\log_6 \sqrt[3]{\frac{(x-1)^{5}}{(x+1)^{10}}}
y
=
lo
g
6
3
(
x
+
1
)
10
(
x
−
1
)
5
Derivatives of Logarithmic Functions: Two Logs
Find
d
y
/
d
x
dy/dx
d
y
/
d
x
for
y
=
5
log
5
(
log
2
t
)
\displaystyle y=5\log_{5}(\log_{2}t)
y
=
5
lo
g
5
(
lo
g
2
t
)
Practice: Log and Ln
Q.
\textbf{Q.}
Q.
Find the derivative of
y
=
log
3
(
7
x
ln
2
)
\displaystyle y=\text{log}_3(7x^{\ln2})
y
=
log
3
(
7
x
l
n
2
)
Derivatives: Logarithmic and Inverse Trigonometric Functions
Evaluate
d
d
x
(
sin
−
1
x
sin
x
log
3
x
)
\displaystyle \frac{\text{d}}{\text{d}x}\left( \frac{\sin^{-1}x\sin x}{\log_3x}\right)
d
x
d
(
lo
g
3
x
sin
−
1
x
sin
x
)
.
Derivatives: Trigonometric and Logarithmic Functions
Find
d
d
y
[
(
y
5
+
ln
y
)
tan
y
]
\displaystyle \frac{\text{d}}{\text{d}y}\left[\left(y^5+\ln y\right)\tan y\right]
d
y
d
[
(
y
5
+
ln
y
)
tan
y
]
.
Compute the derivative of
f
(
x
)
=
log
2
(
x
2
+
2
x
e
x
)
f(x) = \log_2(x^2 + 2xe^x)
f
(
x
)
=
lo
g
2
(
x
2
+
2
x
e
x
)
.
Derivatives of Logarithmic Functions
Evaluate
d
d
x
(
4
x
log
700
x
)
\frac{\text{d}}{\text{d}x}\left(4^x\log_{700}x\right)
d
x
d
(
4
x
lo
g
700
x
)
. It is not necessary to simplify your answer.
Find the derivative of the following function.
f
(
x
)
=
ln
(
x
2
+
x
)
\displaystyle f(x)=\ln{(x^2+\sqrt{x})}
f
(
x
)
=
ln
(
x
2
+
x
)
Given that
f
(
x
)
=
e
cos
(
ln
x
)
f\left(x\right)=e^{\cos\left(\ln x\right)}
f
(
x
)
=
e
c
o
s
(
l
n
x
)
, find
f
′
(
1
)
f'\left(1\right)
f
′
(
1
)
.
Practice: Chain Rule
Find the derivative of
y
=
ln
(
arctan
x
)
y=\ln(\arctan x)
y
=
ln
(
arctan
x
)
Derivatives: Logarithmic Functions
Find the derivative of
f
(
x
)
=
e
x
x
2
+
ln
(
x
)
3
f(x) = e^{x} x^2 + \frac{\ln(x)}{3}
f
(
x
)
=
e
x
x
2
+
3
ln
(
x
)
Derivatives: Logarithmic Functions
Find all the points for which the tangent line of the function
f
(
x
)
=
4
ln
(
x
)
+
x
2
−
3
x
+
5
f(x)= 4 \ln (x) + x^2 - 3x + 5
f
(
x
)
=
4
ln
(
x
)
+
x
2
−
3
x
+
5
has slope equal to 3.
Derivatives of Logarithmic Functions: Normal Lines
Find the equation of the normal line to
f
(
x
)
=
ln
x
at the point
(
e
,
1
)
f(x)=\ln{x} \text{ at the point } (e,1)
f
(
x
)
=
ln
x
at the point
(
e
,
1
)
Note:
This questions requires knowledge of derivatives of logarithms
Derivatives: Logarithmic Functions
Find the equation of the tangent line to the given point for the following function.
f
(
x
)
=
ln
(
x
+
3
x
+
1
)
at
x
=
1
f(x)=\ln\left(\frac{x+3}{x+1}\right) \text{at}\ x=1
f
(
x
)
=
ln
(
x
+
1
x
+
3
)
at
x
=
1
Derivatives: Exponential and Logarithmic Functions
Find the derivative of the following function.
f
(
x
)
=
(
x
2
+
x
ln
x
)
e
x
f(x)=(x^2+x\ln{x})e^{\sqrt{x}}
f
(
x
)
=
(
x
2
+
x
ln
x
)
e
x