Glossary

9.3 Logarithmic differentiation

2 min read•july 22, 2024

is a powerful tool for tackling complex derivatives. It simplifies the process by converting products into sums and powers into coefficients, making it easier to differentiate tricky functions like x^x or products of multiple factors.

This technique is especially handy when dealing with functions that have variable exponents or multiple parts. By taking the natural log of both sides, we can break down complicated expressions into more manageable pieces, saving time and reducing errors in calculations.

Logarithmic Differentiation

Concept of logarithmic differentiation

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  • Technique used to find derivatives of complex functions
    • Particularly useful for functions that are products of several factors ((2x+1)(x2−3)(x)(2x+1)(x^2-3)(\sqrt{x}))
    • Helps with functions involving powers that are functions themselves ((sin⁡(x))cos⁡(x)(\sin(x))^{\cos(x)})
  • Takes the of both sides of an equation before differentiating
    • Based on the properties of logarithms, specifically the : ln⁡(ab)=ln⁡(a)+ln⁡(b)\ln(ab) = \ln(a) + \ln(b)
  • Simplifies the differentiation process by converting products into sums and powers into coefficients
  • Often used when the function is in the form of:
    • f(x)=[g(x)]h(x)f(x) = [g(x)]^{h(x)}, where g(x)g(x) and h(x)h(x) are functions of xx
    • f(x)=u(x)v(x)w(x)...f(x) = u(x)v(x)w(x)..., where u(x)u(x), v(x)v(x), w(x)w(x), etc. are functions of xx

Application for complex functions

  • Steps to apply logarithmic differentiation:
    1. Take the natural logarithm of both sides of the equation
    2. Use the properties of logarithms to simplify the right-hand side
    3. Differentiate both sides of the equation with respect to the independent variable (usually xx)
    4. Solve for the derivative of the original function by multiplying both sides by the original function
  • Example: Find the derivative of f(x)=xxf(x) = x^x using logarithmic differentiation
    1. ln⁡(f(x))=ln⁡(xx)\ln(f(x)) = \ln(x^x)
    2. ln⁡(f(x))=xln⁡(x)\ln(f(x)) = x\ln(x)
    3. f′(x)f(x)=ln⁡(x)+1\frac{f'(x)}{f(x)} = \ln(x) + 1
    4. f′(x)=xx(ln⁡(x)+1)f'(x) = x^x(\ln(x) + 1)

Simplification of derivative calculations

  • Simplifies the differentiation process for functions involving:
    • Products of several factors (f(x)=(2x+1)(x2−3)(x)f(x) = (2x+1)(x^2-3)(\sqrt{x}))
    • Quotients of functions (f(x)=(x+1)3(x−2)2f(x) = \frac{(x+1)^3}{(x-2)^2})
    • Functions raised to the power of another function (f(x)=(sin⁡(x))cos⁡(x)f(x) = (\sin(x))^{\cos(x)})
  • By :
    • Products are converted to sums
    • Quotients are converted to differences
    • Powers become coefficients
  • Makes the differentiation process more manageable and easier to solve

Problem-solving with logarithmic differentiation

  • Can be applied to various problems in calculus and beyond
  • Common applications include:
    • Finding the derivative of a complex function
    • Determining the equation of a tangent line to a curve at a given point
    • Solving optimization problems (maximize the volume of a box with a fixed surface area)
  • When solving problems:
    • Identify the appropriate function to differentiate
    • Apply the steps of logarithmic differentiation correctly
    • Simplify the resulting expression
    • Interpret the results in the context of the problem

Key Terms to Review (17)

Chain Rule: The chain rule is a fundamental technique in calculus used to differentiate composite functions, allowing us to find the derivative of a function that is made up of other functions. This rule is crucial for understanding how different rates of change are interconnected and enables us to tackle complex differentiation problems involving multiple layers of functions.
Change of Base Formula: The change of base formula is a mathematical tool that allows the conversion of logarithms from one base to another. It is expressed as $$\log_b(a) = \frac{\log_k(a)}{\log_k(b)}$$, where \(k\) is any positive number different from 1. This formula is especially useful in situations where calculators or specific logarithmic bases are required for computations, facilitating easier differentiation and integration in problems involving logarithmic functions.
Derivative of ln(x): The derivative of ln(x) is a fundamental concept in calculus, defined as the rate at which the natural logarithm function changes with respect to its variable x. This derivative is crucial for understanding how logarithmic functions behave and is given by the formula $$\frac{d}{dx} \ln(x) = \frac{1}{x}$$ for x > 0. This relationship plays an important role in solving problems involving growth rates, integration, and optimization.
Derivative of log_a(x): The derivative of log_a(x) represents the rate of change of the logarithmic function with base 'a' concerning 'x'. This derivative is essential for understanding how logarithmic functions behave and is commonly expressed using the formula $$\frac{d}{dx} \log_a(x) = \frac{1}{x \ln(a)}$$. It connects to the concept of logarithmic differentiation, which is a powerful technique for finding derivatives of complicated functions by applying properties of logarithms.
Exponential Functions: Exponential functions are mathematical functions of the form $$f(x) = a imes b^{x}$$, where $$a$$ is a constant, $$b$$ is a positive real number, and $$x$$ is the exponent. They describe processes that grow or decay at a constant rate proportional to their current value, making them crucial in modeling real-world phenomena such as population growth and radioactive decay.
Finding slopes of tangent lines: Finding slopes of tangent lines refers to the process of determining the steepness or incline of a curve at a specific point. This involves using the concept of derivatives, where the derivative of a function at a point gives the slope of the tangent line to the curve at that point. This idea is crucial in understanding how functions behave locally and is especially relevant when working with inverse functions and logarithmic differentiation, where these slopes help analyze relationships between variables.
Implicit Differentiation: Implicit differentiation is a technique used to find the derivative of a function defined implicitly, rather than explicitly. In this method, both variables are treated as functions of a third variable, typically 'x', allowing us to differentiate equations that cannot be easily solved for one variable in terms of another. This technique connects closely with the definition of the derivative and is essential when using the chain rule, especially when dealing with equations involving multiple variables or functions that are not isolated.
Logarithmic Differentiation: Logarithmic differentiation is a technique used to differentiate functions by taking the natural logarithm of both sides, simplifying the differentiation process, especially for products, quotients, or power functions. This method takes advantage of the properties of logarithms to transform complicated expressions into simpler ones, making it easier to apply the rules of differentiation. It is particularly useful when dealing with functions that are products or quotients of other functions or when the function has variables raised to variable powers.
Logarithmic identity: A logarithmic identity is a mathematical statement that expresses the relationship between logarithms and their arguments in a way that holds true for all valid values. These identities are essential in simplifying logarithmic expressions and solving equations, particularly in calculus when differentiating complex functions. Logarithmic identities play a critical role in logarithmic differentiation, enabling students to rewrite complicated functions in simpler forms for easier differentiation.
Maximizing/minimizing functions: Maximizing and minimizing functions refers to the process of finding the highest or lowest values of a function within a given domain. This is essential in various applications, such as economics, engineering, and natural sciences, where optimizing a certain quantity can lead to better outcomes. The key concept often involves analyzing critical points, where the derivative of the function equals zero or is undefined, to determine local maxima or minima.
Misapplying the product rule: Misapplying the product rule occurs when one incorrectly uses the product rule for differentiation, leading to errors in calculating the derivative of a product of two functions. This mistake often arises from confusion about the correct application of the rule, which states that if you have two functions multiplied together, their derivative is not simply the product of their individual derivatives but instead requires additional terms. Recognizing and avoiding this error is crucial for accurate differentiation, especially when dealing with more complex expressions.
Natural logarithm: The natural logarithm is the logarithm to the base e, where e is an irrational constant approximately equal to 2.71828. It is commonly denoted as ln(x) and is used in many mathematical contexts, particularly in calculus and exponential growth scenarios. The natural logarithm has unique properties that make it particularly useful for solving equations involving exponential functions and in the process of differentiation.
Neglecting to differentiate both sides: Neglecting to differentiate both sides refers to the oversight of not applying differentiation equally to both sides of an equation during the process of solving or manipulating that equation. This concept highlights the importance of maintaining mathematical integrity and ensuring that any operation applied to one side of an equation is also applied to the other side, especially when dealing with functions that require logarithmic differentiation.
Polynomial Functions: Polynomial functions are mathematical expressions that represent relationships involving variables raised to whole number powers, where the coefficients can be real or complex numbers. They are continuous and smooth across their domain, making them crucial in calculus for understanding derivatives, integrals, and behavior of functions.
Product Rule: The product rule is a fundamental principle in calculus that provides a method for finding the derivative of the product of two functions. It states that if you have two functions, say $$u(x)$$ and $$v(x)$$, the derivative of their product can be calculated using the formula: $$ (uv)' = u'v + uv' $$, where $$u'$$ and $$v'$$ are the derivatives of $$u$$ and $$v$$ respectively. This concept is crucial in understanding how derivatives work when dealing with more complex functions that are products of simpler ones.
Quotient Rule: The quotient rule is a method for finding the derivative of a function that is the ratio of two differentiable functions. It states that if you have a function that can be expressed as $$f(x) = \frac{g(x)}{h(x)}$$, where both $$g$$ and $$h$$ are differentiable, then the derivative is given by $$f'(x) = \frac{g'(x)h(x) - g(x)h'(x)}{(h(x))^2}$$. This rule connects to understanding how rates of change behave in division scenarios, as well as its application alongside other rules such as the product rule and logarithmic differentiation.
Taking the logarithm of both sides: Taking the logarithm of both sides is a mathematical technique used to simplify equations, particularly when dealing with exponential functions. This method allows for easier differentiation and manipulation of the equation by transforming multiplicative relationships into additive ones, making it especially useful in finding derivatives of complex functions.
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