Limits Derivatives Integrals Infinite Series Parametrics Polar Coordinates Conics
Limits
Epsilon-Delta Definition
Finite Limits
One-Sided Limits
Infinite Limits
Trig Limits
Pinching Theorem
Indeterminate Forms
L'Hopitals Rule
Limits That Do Not Exist
Continuity & Discontinuities
Intermediate Value Theorem
Derivatives
Power Rule
Product Rule
Quotient Rule
Chain Rule
Trig and Inverse Trig
Implicit Differentiation
Exponentials & Logarithms
Logarithmic Differentiation
Hyperbolic Functions
Higher Order Derivatives
Differentials
Slope, Tangent, Normal...
Linear Motion
Mean Value Theorem
Graphing
1st Deriv, Critical Points
2nd Deriv, Inflection Points
Related Rates Basics
Related Rates Areas
Related Rates Distances
Related Rates Volumes
Optimization
Integrals
Definite Integrals
Integration by Substitution
Integration By Parts
Partial Fractions
Improper Integrals
Basic Trig Integration
Sine/Cosine Integration
Secant/Tangent Integration
Trig Integration Practice
Trig Substitution
Linear Motion
Area Under/Between Curves
Volume of Revolution
Arc Length
Surface Area
Work
Moments, Center of Mass
Exponential Growth/Decay
Laplace Transforms
Describing Plane Regions
Infinite Series
Divergence (nth-Term) Test
p-Series
Geometric Series
Alternating Series
Telescoping Series
Ratio Test
Limit Comparison Test
Direct Comparison Test
Integral Test
Root Test
Absolute Convergence
Conditional Convergence
Power Series
Taylor/Maclaurin Series
Interval of Convergence
Remainder & Error Bounds
Fourier Series
Study Techniques
Choosing A Test
Sequences
Infinite Series Table
Practice Problems
Exam Preparation
Exam List
Parametrics
Parametric Curves
Parametric Surfaces
Slope & Tangent Lines
Area
Arc Length
Surface Area
Volume
Polar Coordinates
Converting
Slope & Tangent Lines
Area
Arc Length
Surface Area
Conics
Parabolas
Ellipses
Hyperbolas
Conics in Polar Form
Vectors Vector Functions Partial Derivatives/Integrals Vector Fields Laplace Transforms Tools
Vectors
Unit Vectors
Dot Product
Cross Product
Lines In 3-Space
Planes In 3-Space
Lines & Planes Applications
Angle Between Vectors
Direction Cosines/Angles
Vector Projections
Work
Triple Scalar Product
Triple Vector Product
Vector Functions
Projectile Motion
Unit Tangent Vector
Principal Unit Normal Vector
Acceleration Vector
Arc Length
Arc Length Parameter
Curvature
Vector Functions Equations
MVC Practice Exam A1
Partial Derivatives
Directional Derivatives
Lagrange Multipliers
Tangent Plane
MVC Practice Exam A2
Partial Integrals
Describing Plane Regions
Double Integrals-Rectangular
Double Integrals-Applications
Double Integrals-Polar
Triple Integrals-Rectangular
Triple Integrals-Cylindrical
Triple Integrals-Spherical
MVC Practice Exam A3
Vector Fields
Curl
Divergence
Conservative Vector Fields
Potential Functions
Parametric Curves
Line Integrals
Green's Theorem
Parametric Surfaces
Surface Integrals
Stokes' Theorem
Divergence Theorem
MVC Practice Exam A4
Laplace Transforms
Unit Step Function
Unit Impulse Function
Square Wave
Shifting Theorems
Solve Initial Value Problems
Prepare For Calculus 1
Ready For Calculus 2?
Trig Formulas
Describing Plane Regions
Parametric Curves
Linear Algebra Review
Word Problems
Mathematical Logic
Calculus Notation
Simplifying
Practice Exams
More Math Help
Tutoring
Tools and Resources
Learning/Study Techniques
Math/Science Learning
Memorize To Learn
Music and Learning
Note-Taking
Motivation
Instructor or Coach?
Books
Math Books
How To Read Math Books

You CAN Ace Calculus

17calculus > derivatives > logarithmic differentiation

Calculus Main Topics

Derivatives

Derivative Applications

Optimization

Single Variable Calculus

Multi-Variable Calculus

Tools

math tools

general learning tools

Related Topics and Links

Logarithmic Differentiation

This topic is usually found in the section discussing implicit differentiation and sometimes instructors do not make a distinction between the two. But logarithmic differentiation is a very specific technique and often uses implicit differentiation along the way.

There are two main types of equations that you will use logarithmic differentiation on
1. equations where you have a variable in an exponent
2. equations that are quite complicated and can be simplified using logarithms.
In both cases, we introduce logarithms into the equation that may not have been there before, apply some simple rules and then take the derivative. Let's look at each case.

Variables In The Exponent

Remember that you can use the power rule on $$x^2$$ but you can't use the power rule on $$2^x$$ or $$y^x$$. [ why? ] So, what we do is introduce a natural log into the equation, without changing the problem of course. The goal is to bring the exponent down so that we can take the derivative of it.

These are the rules we use

1. $$\ln(x^y) = y~ \ln(x)$$

2. $$e^{\ln(z)} = z$$

Notice the first rule brings the exponent down in front of the natural log term, in which case, we can use the product rule to take the derivative. The second rule is usually used to reverse the process after taking the derivative.

Let's look at an example. One of the practice problems shows how to calculate the derivative of $$y = x^x$$. To do this one, we need to bring the x in the exponent down (since we can't use the power rule). To accomplish this, we take the natural log of both sides, like this:

$$\displaystyle{ \begin{array}{rcl} y & =& x^x \\ \ln(y) & = & \ln(x^x) \\ \ln(y) & = & x \ln(x) \end{array} }$$

Now we can take the derivative of this last equation using chain rule and the product rule. [ For a complete solution, see practice problem A01. ]

Simplifying

In some cases, we could use the product and/or quotient rules to take a derivative but, using logarithmic differentiation, the derivative would be much easier to find. One of the practice problems is to take the derivative of $$\displaystyle{ y = \frac{(\sin(x))^2(x^3+1)^4}{(x+3)^8} }$$. We could use the product and quotient rules here but, if we take the logarithm of both sides, simplify, take the derivative, then convert back, it is much easier. [ see practice problem B01 for how to find the derivative $$dy/dx$$ ]

Search 17Calculus

Practice Problems

Instructions - - Unless otherwise instructed, calculate $$dy/dx$$ of the following equations using logarithmic differentiation. Give your final answers in exact, completely factored form.

 Level A - Basic

Practice A01

$$\displaystyle{y=x^x}$$

solution

Practice A02

$$\displaystyle{y=x^{e^x}}$$

solution

Practice A03

$$\displaystyle{y=(\ln x)^x}$$

solution

Practice A04

$$\displaystyle{y=(x^2)^{\sin(x)}}$$

solution

Practice A05

$$\displaystyle{y=5^x}$$

solution

Practice A06

$$\displaystyle{y = x^{\sin(x)}}$$

solution

Practice A07

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

solution

Practice A08

for $$\displaystyle{f(x)=x^{2x}}$$, find $$f'(x)$$ and $$f'(1)$$

solution

Practice A09

$$\displaystyle{y=\frac{x^4\sqrt{x-3}}{(x+1)^3}}$$

solution

Practice A10

$$\displaystyle{y=\sqrt{\frac{1-x}{1+x}}}$$

solution

Practice A11

$$\displaystyle{y=x^{x^2}}$$

solution

Practice A12

$$\displaystyle{y=(\sin(x))^x}$$

solution

Practice A13

$$\displaystyle{y=(\cos x)^{\tan x}}$$

solution

Practice A14

$$\displaystyle{y=\frac{2(x^2+1)}{\sqrt{\cos(2x)}}}$$

solution

Practice A15

$$\displaystyle{y=(\sin \theta)^{\sqrt{\theta}}}$$

solution

Practice A16

$$\displaystyle{f(x)=(2x-3)^2(5x^2+2)^3}$$

solution

 Level B - Intermediate

Practice B01

$$\displaystyle{y=\frac{(\sin(x))^2(x^3+1)^4}{(x+3)^8}}$$

solution

Practice B02

$$\displaystyle{y=\sqrt{x}e^{x^2}(x^2+1)^{10}}$$

solution

Practice B03

for $$\displaystyle{f(x)=\frac{x^2(x+2)^4}{(2x^2-1)^3}}$$, find $$f'(x)$$ and $$f'(4)$$

solution

Practice B04

$$\displaystyle{y=\sqrt[3]{\frac{x(x^2+1)^4}{x^3-2}}}$$

solution

 Level C - Advanced

Practice C01

$$\displaystyle{y=x^{(x^x)}}$$

solution

4