You CAN Ace Calculus

 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.

### 17Calculus Subjects Listed Alphabetically

Single Variable Calculus

 Absolute Convergence Alternating Series Arc Length Area Under Curves Chain Rule Concavity Conics Conics in Polar Form Conditional Convergence Continuity & Discontinuities Convolution, Laplace Transforms Cosine/Sine Integration Critical Points Cylinder-Shell Method - Volume Integrals Definite Integrals Derivatives Differentials Direct Comparison Test Divergence (nth-Term) Test
 Ellipses (Rectangular Conics) Epsilon-Delta Limit Definition Exponential Derivatives Exponential Growth/Decay Finite Limits First Derivative First Derivative Test Formal Limit Definition Fourier Series Geometric Series Graphing Higher Order Derivatives Hyperbolas (Rectangular Conics) Hyperbolic Derivatives
 Implicit Differentiation Improper Integrals Indeterminate Forms Infinite Limits Infinite Series Infinite Series Table Infinite Series Study Techniques Infinite Series, Choosing a Test Infinite Series Exam Preparation Infinite Series Exam A Inflection Points Initial Value Problems, Laplace Transforms Integral Test Integrals Integration by Partial Fractions Integration By Parts Integration By Substitution Intermediate Value Theorem Interval of Convergence Inverse Function Derivatives Inverse Hyperbolic Derivatives Inverse Trig Derivatives
 Laplace Transforms L'Hôpital's Rule Limit Comparison Test Limits Linear Motion Logarithm Derivatives Logarithmic Differentiation Moments, Center of Mass Mean Value Theorem Normal Lines One-Sided Limits Optimization
 p-Series Parabolas (Rectangular Conics) Parabolas (Polar Conics) Parametric Equations Parametric Curves Parametric Surfaces Pinching Theorem Polar Coordinates Plane Regions, Describing Power Rule Power Series Product Rule
 Quotient Rule Radius of Convergence Ratio Test Related Rates Related Rates Areas Related Rates Distances Related Rates Volumes Remainder & Error Bounds Root Test Secant/Tangent Integration Second Derivative Second Derivative Test Shifting Theorems Sine/Cosine Integration Slope and Tangent Lines Square Wave Surface Area
 Tangent/Secant Integration Taylor/Maclaurin Series Telescoping Series Trig Derivatives Trig Integration Trig Limits Trig Substitution Unit Step Function Unit Impulse Function Volume Integrals Washer-Disc Method - Volume Integrals Work

Multi-Variable Calculus

 Acceleration Vector Arc Length (Vector Functions) Arc Length Function Arc Length Parameter Conservative Vector Fields Cross Product Curl Curvature Cylindrical Coordinates
 Directional Derivatives Divergence (Vector Fields) Divergence Theorem Dot Product Double Integrals - Area & Volume Double Integrals - Polar Coordinates Double Integrals - Rectangular Gradients Green's Theorem
 Lagrange Multipliers Line Integrals Partial Derivatives Partial Integrals Path Integrals Potential Functions Principal Unit Normal Vector
 Spherical Coordinates Stokes' Theorem Surface Integrals Tangent Planes Triple Integrals - Cylindrical Triple Integrals - Rectangular Triple Integrals - Spherical
 Unit Tangent Vector Unit Vectors Vector Fields Vectors Vector Functions Vector Functions Equations

Differential Equations

 Boundary Value Problems Bernoulli Equation Cauchy-Euler Equation Chebyshev's Equation Chemical Concentration Classify Differential Equations Differential Equations Euler's Method Exact Equations Existence and Uniqueness Exponential Growth/Decay
 First Order, Linear Fluids, Mixing Fourier Series Inhomogeneous ODE's Integrating Factors, Exact Integrating Factors, Linear Laplace Transforms, Solve Initial Value Problems Linear, First Order Linear, Second Order Linear Systems
 Partial Differential Equations Polynomial Coefficients Population Dynamics Projectile Motion Reduction of Order Resonance
 Second Order, Linear Separation of Variables Slope Fields Stability Substitution Undetermined Coefficients Variation of Parameters Vibration Wronskian

### Search Practice Problems

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.

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.

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.

GOT IT. THANKS!

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

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

Problem Statement

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

Solution

### 1085 solution video

video by PatrickJMT

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

Problem Statement

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

Solution

### 1091 solution video

video by PatrickJMT

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

Problem Statement

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

Solution

### 1080 solution video

video by PatrickJMT

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

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

Problem Statement

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

Solution

### 1088 solution video

video by PatrickJMT

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

Problem Statement

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

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

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.

 $$\displaystyle{ \frac{d}{dx}[ \ln(3x^2+9x-5) ] }$$ $$\displaystyle{ \frac{d}{du}[\ln(u)] \cdot \frac{d}{dx}[3x^2+9x-5] }$$ $$\displaystyle{ \frac{1}{u} \cdot (6x+9) }$$ $$\displaystyle{ \frac{1}{3x^2+9x-5} \cdot [ 3(2x+3) ] }$$

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

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

Problem Statement

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

Solution

### 1081 solution video

video by PatrickJMT

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

Problem Statement

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

Solution

### 1156 solution video

video by MathTV

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

Problem Statement

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

Solution

### 1082 solution video

video by PatrickJMT

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

Problem Statement

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

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

Problem Statement

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

Solution

### 1070 solution video

video by Krista King Math

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

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

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

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.

 $$f^{(3)}(x) = \displaystyle{\frac{d}{dx}\left[ -9(2+3x)^{-2} \right]}$$ $$\displaystyle{-9(-2)(2+3x)^{-3}\frac{d}{dx}[2+3x]}$$ $$18(2+3x)^{-3} (3)$$ $$\displaystyle{\frac{54}{(2+3x)^3}}$$

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

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

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

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

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

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

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

Problem Statement

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

Solution

### 1089 solution video

video by PatrickJMT

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

Problem Statement

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

Solution

### 1090 solution video

video by PatrickJMT

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

Problem Statement

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

Solution

### 1087 solution video

video by PatrickJMT

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

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