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Topics You Need To Understand For This Page

basic derivative rules

power rule

product rule

quotient rule

For the basic logarithm derivatives you do not need the chain rule. But we discuss it on this page. Each section is labeled. So if you have not studied the chain rule yet, you can read the sections that apply to you and then come back here once you have studied it.

Related Topics and Links

17Calculus Subjects Listed Alphabetically

Single Variable Calculus

Multi-Variable Calculus

Differential Equations

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Do you have a practice problem number but do not know on which page it is found? If so, enter the number below and click 'page' to go to the page on which it is found or click 'practice' to be taken to the practice problem.

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Logarithm Derivative

The rule is not as simple as the exponential derivative but it is still very straightforward.

Basic Logarithm Rule

\(\displaystyle{ \frac{d}{dt}[\ln(t)] = \frac{1}{t} }\)

Logarithm With Chain Rule

\(\displaystyle{ \frac{d}{dt}[\ln(u)] = \frac{1}{u}\frac{du}{dt} }\)

It is probably not clear just from the equation that the derivative of \(\ln(x)\) is \(1/x\). Here is a great video explaining, first intuitively, then from the limit, where this derivative comes from. Although he says you can stop the video after the intuitive explanation, watching the entire video will help you a lot (it's not very long).

MathTV - The Derivative of the Natural Log Function [12min-15secs]

video by MathTV

Many times, it helps to simplify a logarithmic expression before taking the derivative. Here are a few rules that should help you.

\( \ln(xy) = \ln(x) + \ln(y) \)

\( \ln(x/y) = \ln(x) - \ln(y) \)

\( \ln(x^y) = y \ln(x) \)

review logarithms

Here is a short video clip that goes through these equations again.

PatrickJMT - Derivatives of Logarithmic Functions and Examples [1min-29secs]

video by PatrickJMT

Before jumping into some practice problems, take a couple of minutes to watch this next video. It will help you see some common mistakes that you can avoid when taking the derivative of logarithm functions.

MathTV - Common Mistakes for Natural Logs and the Chain Rule [3min-43secs]

video by MathTV

Practice

Conversion Between A-B-C Level (or 1-2-3) and New Numbered Practice Problems

Please note that with this new version of 17calculus, the practice problems have been relabeled but they are MOSTLY in the same order. Here is a list converting the old numbering system to the new.

Logarithm Derivatives - Practice Problems Conversion

[13-1071] - [14-1085] - [15-1084] - [16-1088] - [17-975] - [18-1081] - [19-1156] - [20-1082] - [21-1070]

[22-1072] - [23-1080] - [24-976] - [25-1091] - [26-1073] - [27-1086] - [28-1089] - [29-1090] - [30-1087] - [31-1083]

Note: In the previous version, these practice problems were on the exponential derivatives page. For this new format, they were moved to this new page.

Please update your notes to this new numbering system. The display of this conversion information is temporary.

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Unless otherwise instructed, calculate the derivative of these functions. Here are a few practice problems that do not require the chain rule.

\(\displaystyle{f(x)=\frac{1}{\ln(x)}}\)

Problem Statement

Calculate the derivative of \(\displaystyle{f(x)=\frac{1}{\ln(x)}}\).

Solution

1071 solution video

video by Krista King Math

close solution

\(f(x)=\sqrt{x}\ln(x)\)

Problem Statement

Calculate the derivative of \(f(x)=\sqrt{x}\ln(x)\).

Solution

1085 solution video

video by PatrickJMT

close solution

\(\displaystyle{y=\frac{\ln x}{1+\ln(2x)}}\)

Problem Statement

Calculate the derivative of \(\displaystyle{y=\frac{\ln x}{1+\ln(2x)}}\).

Solution

1091 solution video

video by PatrickJMT

close solution

\(y=\ln(x)+2^x+\sin(x)\)

Problem Statement

Calculate the derivative of \(y=\ln(x)+2^x+\sin(x)\).

Solution

1080 solution video

video by PatrickJMT

close solution

These practice problems require the chain rule. As before calculate the derivative of these functions, unless otherwise instructed.

Basic Problems

\(f(x)=\ln(x^2+10)\)

Problem Statement

Calculate the derivative of \(f(x)=\ln(x^2+10)\).

Solution

1084 solution video

video by PatrickJMT

close solution

\(y=\ln(x^2+x)\)

Problem Statement

Calculate the derivative of \(y=\ln(x^2+x)\).

Solution

1088 solution video

video by PatrickJMT

close solution

\(\displaystyle{ \ln(3x^2+9x-5) }\)

Problem Statement

Use the chain rule to calculate the derivative of \(\displaystyle{ \ln(3x^2+9x-5) }\).

Final Answer

\(\displaystyle{ \frac{3(2x+3)}{3x^2+9x-5} }\)

Problem Statement

Use the chain rule to calculate the derivative of \(\displaystyle{ \ln(3x^2+9x-5) }\).

Solution

\(\displaystyle{ \frac{d}{dx}[ \ln(3x^2+9x-5) ] }\)

Apply the chain rule.

\(\displaystyle{ \frac{1}{3x^2+9x-5}\cdot \frac{d}{dx}[3x^2+9x-5] }\)

\(\displaystyle{ \frac{1}{3x^2+9x-5}\cdot (6x+9) }\)

\(\displaystyle{ \frac{3(2x+3)}{3x^2+9x-5} }\)

Using the substitution \( u = 3x^2+9x-5 \) this could have solved more explicitly.

\(\displaystyle{ \frac{d}{dx}[ \ln(3x^2+9x-5) ] }\)

\(\displaystyle{ \frac{d}{du}[\ln(u)] \cdot \frac{d}{dx}[3x^2+9x-5] }\)

\(\displaystyle{ \frac{1}{u} \cdot (6x+9) }\)

\(\displaystyle{ \frac{1}{3x^2+9x-5} \cdot [ 3(2x+3) ] }\)

Final Answer

\(\displaystyle{ \frac{3(2x+3)}{3x^2+9x-5} }\)

close solution

\(h(x)=\ln(x^2+3x+4)\)

Problem Statement

Calculate the derivative of \(h(x)=\ln(x^2+3x+4)\).

Solution

1081 solution video

video by PatrickJMT

close solution

\(f(x)=\ln(5x^2+2x-7)\)

Problem Statement

Calculate the derivative of \(f(x)=\ln(5x^2+2x-7)\).

Solution

1156 solution video

video by MathTV

close solution

\(g(x)=\log_4(x^3+8x)\)

Problem Statement

Calculate the derivative of \(g(x)=\log_4(x^3+8x)\).

Solution

1082 solution video

video by PatrickJMT

close solution

\(\displaystyle{f(x)=\ln\sqrt[3]{x^3-x}}\)

Problem Statement

Calculate the derivative of \(\displaystyle{f(x)=\ln\sqrt[3]{x^3-x}}\) .

Final Answer

\(\displaystyle{ \frac{3x^2-1}{3x(x^2-1)} }\)

Problem Statement

Calculate the derivative of \(\displaystyle{f(x)=\ln\sqrt[3]{x^3-x}}\) .

Solution

1070 solution video

video by Krista King Math

Final Answer

\(\displaystyle{ \frac{3x^2-1}{3x(x^2-1)} }\)

close solution

\(\displaystyle{f(x)=\ln\left(x\sqrt{x^2+1}\right)}\)

Problem Statement

Calculate the derivative of \(\displaystyle{f(x)=\ln\left(x\sqrt{x^2+1}\right)}\).

Solution

1072 solution video

video by Krista King Math

close solution

Calculate the first three derivatives of \(\displaystyle{ f(x)=\ln(2+3x) }\).

Problem Statement

Calculate the first three derivatives of \(\displaystyle{ f(x)=\ln(2+3x) }\).

Solution

First Derivative

\(\displaystyle{ f'(x) = \frac{d}{dx}[ \ln(2+3x) ] }\)

\(\displaystyle{\frac{1}{2+3x} \frac{d}{dx}[2+3x]}\)

\(\displaystyle{\frac{1}{2+3x} (3) = \frac{3}{2+3x}}\)

Second Derivative
We can use the either quotient rule or product rule here. The product rule is the easiest here because of the constant in the numerator but we need to rewrite the first derivative as \(\displaystyle{ \frac{3}{2+3x} = 3(2+3x)^{-1} }\).

\(\displaystyle{ f''(x) = \frac{d}{dx} \left[ 3(2+3x)^{-1} \right] }\)

\( \displaystyle{3(-1)(2+3x)^{-2} \frac{d}{dx}[2+3x] }\)

\(\displaystyle{-3(2+3x)^{-2} (3) = \frac{-9}{(2+3x)^{2}} }\)

Third Derivative
Again, we can use either the quotient rule or product rule and we choose the product rule.

\( f^{(3)}(x) = \displaystyle{\frac{d}{dx}\left[ -9(2+3x)^{-2} \right]} \)

\( \displaystyle{-9(-2)(2+3x)^{-3}\frac{d}{dx}[2+3x]} \)

\( 18(2+3x)^{-3} (3) \)

\(\displaystyle{\frac{54}{(2+3x)^3}}\)

Final Answers

\(\displaystyle{f'(x) =\frac{3}{2+3x}}\)

\(\displaystyle{f''(x)=\frac{-9}{(2+3x)^{2}}}\)

\(\displaystyle{f^{(3)}(x)=\frac{54}{(2+3x)^3}}\)

close solution

Intermediate Problems

\(\displaystyle{f(x)=\ln\left[\frac{(2x+1)^5}{\sqrt{x^2+1}}\right]}\)

Problem Statement

Calculate the derivative of \(\displaystyle{f(x)=\ln\left[\frac{(2x+1)^5}{\sqrt{x^2+1}}\right]}\).

Solution

1073 solution video

video by Krista King Math

close solution

\(\displaystyle{f(x)=\ln\left[\frac{(2x+1)^3}{(3x-1)^4}\right]}\)

Problem Statement

Calculate the derivative of \(\displaystyle{f(x)=\ln\left[\frac{(2x+1)^3}{(3x-1)^4}\right]}\).

Solution

1086 solution video

video by PatrickJMT

close solution

\(y=\sqrt[3]{\log_7(x)}\)

Problem Statement

Calculate the derivative of \(y=\sqrt[3]{\log_7(x)}\).

Solution

1089 solution video

video by PatrickJMT

close solution

\(y=\ln(x^4\sin x)\)

Problem Statement

Calculate the derivative of \(y=\ln(x^4\sin x)\).

Solution

1090 solution video

video by PatrickJMT

close solution

\(y=[\log_4(1+e^x)]^2\)

Problem Statement

Calculate the derivative of \(y=[\log_4(1+e^x)]^2\).

Solution

1087 solution video

video by PatrickJMT

close solution

\(p(x)=\ln\left[ x^2 \cdot \sqrt{x^3+3x} \cdot (x+2)^4 \right] \)

Problem Statement

Calculate the derivative of \(p(x)=\ln\left[ x^2 \cdot \sqrt{x^3+3x} \cdot (x+2)^4 \right] \).

Solution

1083 solution video

video by PatrickJMT

close solution
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