Introduction
Precalculus
Getting Started
Calculus 1
1. Limits
3. Derivatives
5. Applications of Derivatives
6. 1. Increasing and Decreasing Function
  2. Minimum and Maximum Values
7. Integrals
8. 1. Indefinite Integrals
  2. Computing Indefinite Integrals
  3. Substitution Rule for Indefinite Integrals
  4. Area Problem
Calculus 2
1. Integration Techniques
2. 1. Integration by Parts
  2. Integrals Involving Trig Functions
  3. Trig Substitutions
  4. Partial Fractions
  5. Integrals Involving Roots
  6. Integrals Involving Quadratics
3. Parametric Equations and Polar Coordinates
4. 1. Differentiation Formulas
  2. Product and Quotient Rule
  3. Chain Rule
  4. Implicit Differentiation
  5. Higher Order Derivatives
  6. Logarithmic Differentiation
5. Series and Sequences
6. 1. Indefinite Integrals
  2. Computing Indefinite Integrals
  3. Substitution Rule for Indefinite Integrals
  4. Area Problem
7. Vectors
8. 3-Dimensional Space
Calculus 3
1. LimitThree Dimensional Coordinate Systems
2. 1. Equations of Lines
  2. Equations of Planes
  3. Quadratic Surfaces
  4. Functions of Multiple Variables
3. Partial Derivatives
4. 1. Higher Order Partial Derivatives
  2. Differentials
  3. Chain Rule
  4. DirectioChain Rulenal Derivatives
  5. Gradient
5. Multiple Integrals
6. 1. Iterated Integrals in Cylindrical Coordinates
  2. Double Integrals
  3. Double Integrals in Polar Coordinates
  4. Triple Integrals in Spherical Coordinates
  5. Change of Variables
  6. Surface Area
7. Line Integrals
8. 1. Line Integrals With Respect to Arc Length
  2. Double Integrals in Polar Coordinates
  3. Line Integrals With Respect to x and y
  4. Line Integrals of Vector Fields
  5. Fundamental Theorem of Line Integrals
  6. Conservative Vector Fields
  7. Potential Functions
  8. Green's Theorem
9. Surface Integrals
10. 1. Parametric Surfaces
  2. Surface Integrals
  3. Surface Integrals of Vector Fields
  4. Stokes' Theorem
  5. Divergence Theorem
Cheatsheet

Logarithmic Differenation

What is Logarithmic Differenation?

Logarithmic differentiation is a technique that allows us to differentiate functions that cannot be directly differentiated using the standard rules of calculus. It is based on the property that the derivative of the logarithm of a function is equal to the derivative of the function divided by the function itself.

Let's say we want to differentiate the function $f(x) = x^3$. Using the power rule for derivatives, we can write the derivative as:

$$\frac{d}{dx} x^3 = 3x^2$$

Now, let's say we want to differentiate the function $g(x) = \ln(x^3)$. We can use logarithmic differentiation to find the derivative of $g(x)$ as follows:

$$\frac{d}{dx} \ln(x^3) = \frac{1}{x^3} \cdot \frac{d}{dx} x^3$$

Using the power rule, we can simplify this expression to:

$$\frac{d}{dx} \ln(x^3) = \frac{3x^2}{x^3} = \frac{3}{x}$$

So, the derivative of $\ln(x^3)$ is $\frac{3}{x}$. Note that this is the same as the derivative of $f(x)$ divided by $f(x)$ itself:

$$\frac{\frac{d}{dx} x^3}{x^3} = \frac{3x^2}{x^3} = \frac{3}{x}$$

This is the basic idea behind logarithmic differentiation. It allows us to differentiate functions that cannot be directly differentiated using the standard rules of calculus by expressing them in terms of logarithms and applying the property of the derivative of the logarithm.