∬Differential Calculus Unit 9 – Inverse Function & Logarithm Derivatives

Key Concepts

• Inverse functions reverse the input and output of the original function
• Logarithmic functions are the inverse of exponential functions
  • $\log_b(x)$ is the power to which $b$ must be raised to get $x$
• The derivative of an inverse function is related to the derivative of the original function
  • Involves reciprocals and the chain rule
• Logarithm derivatives utilize the properties of logarithms and the chain rule
  • Logarithmic differentiation is useful for products, quotients, and powers
• Applications include solving equations, modeling growth, and analyzing rates of change
• Common mistakes involve incorrect use of properties or forgetting to apply the chain rule
• Practice problems reinforce understanding and develop problem-solving skills

Inverse Functions Basics

• An inverse function, denoted as $f^{-1}(x)$, "undoes" the original function $f(x)$
  • If $f(a) = b$, then $f^{-1}(b) = a$
• To find the inverse function, swap $x$ and $y$ in the original function and solve for $y$
  • Example: If $f(x) = 2x + 1$, then $f^{-1}(x) = \frac{x-1}{2}$
• The graphs of a function and its inverse are reflections across the line $y = x$
• A function must be one-to-one (injective) to have an inverse
  • Horizontal line test: If any horizontal line intersects the graph more than once, the function is not one-to-one
• The domain of a function becomes the range of its inverse, and vice versa
• Composition of a function with its inverse results in the identity function
  • $f(f^{-1}(x)) = f^{-1}(f(x)) = x$

Logarithmic Functions

• Logarithmic functions are the inverse of exponential functions
  • If $y = b^x$, then $x = \log_b(y)$, where $b$ is the base
• Common logarithmic bases include 10 (common log), $e$ (natural log), and 2 (binary log)
• Properties of logarithms:
  • $\log_b(xy) = \log_b(x) + \log_b(y)$
  • $\log_b(\frac{x}{y}) = \log_b(x) - \log_b(y)$
  • $\log_b(x^n) = n\log_b(x)$
  • $\log_b(1) = 0$ and $\log_b(b) = 1$
• Change of base formula: $\log_b(x) = \frac{\log_a(x)}{\log_a(b)}$
• Logarithmic functions are continuous and increasing for bases greater than 1
• The graph of a logarithmic function has a vertical asymptote at $x = 0$ and grows slowly for large $x$

Derivative Rules for Inverse Functions

• The derivative of an inverse function is related to the derivative of the original function
  • If $y = f^{-1}(x)$, then $\frac{dy}{dx} = \frac{1}{f'(y)}$
• To find the derivative of an inverse function:
  1. Replace $x$ with $y$ in the original function
  2. Differentiate the original function with respect to $y$
  3. Solve for $\frac{dy}{dx}$ by taking the reciprocal
  4. Replace $y$ with $f^{-1}(x)$
• The chain rule is often necessary when applying the inverse function derivative rule
  • Example: If $y = (2x + 1)^3$, then $\frac{dy}{dx} = \frac{1}{6(2y^{1/3} - 1)}$
• Inverse trigonometric functions (arcsin, arccos, arctan) have specific derivative rules
  • Example: $\frac{d}{dx} \arcsin(x) = \frac{1}{\sqrt{1-x^2}}$

Logarithm Derivative Techniques

• Logarithmic differentiation is a technique for finding derivatives of complicated functions
  • Involves taking the natural log of both sides of an equation and using properties of logarithms
• Useful for differentiating products, quotients, and functions raised to variable powers
• Steps for logarithmic differentiation:
  1. Take the natural log of both sides of the equation
  2. Use logarithm properties to simplify the right-hand side
  3. Differentiate both sides with respect to $x$ using the chain rule
  4. Solve for $\frac{dy}{dx}$ by multiplying both sides by $y$
• Example: To find $\frac{d}{dx} (x^x)$:
  1. $\ln(y) = \ln(x^x) = x\ln(x)$
  2. $\frac{1}{y} \frac{dy}{dx} = \ln(x) + 1$
  3. $\frac{dy}{dx} = x^x(\ln(x) + 1)$
• Logarithmic differentiation is also helpful for differentiating functions with multiple factors
  • Example: $y = (2x + 1)^3 (3x - 2)^2$

Applications and Examples

• Inverse functions are used to solve equations and model real-world situations
  • Example: If a car's value depreciates according to $V(t) = 20000(0.8)^t$, find when the value will be $5000
• Logarithmic functions model exponential growth and decay
  • Example: Population growth, radioactive decay, compound interest
• Logarithmic differentiation is applied in physics, economics, and engineering
  • Example: Analyzing the elasticity of demand in economics
• Inverse trigonometric functions appear in calculus, physics, and geometry
  • Example: Calculating the angle of elevation in a right triangle
• Derivatives of inverse functions and logarithms are used in optimization problems
  • Example: Minimizing the cost of production in a manufacturing process
• Understanding these concepts is crucial for advanced mathematics and science courses
  • Example: Differential equations, complex analysis, and mathematical modeling

Common Mistakes and Tips

• Forgetting to use the chain rule when differentiating inverse or logarithmic functions
  • Tip: Identify the "inner" and "outer" functions and apply the chain rule
• Incorrectly applying logarithm properties or confusing ln and log
  • Tip: Practice using properties and remember ln is base e, log is base 10
• Misinterpreting the domain and range of inverse functions
  • Tip: Sketch the graph of the original function to identify the domain and range
• Neglecting to simplify expressions or cancel terms when possible
  • Tip: Look for common factors or logarithms that can be combined
• Making algebraic errors when solving for the inverse function or derivative
  • Tip: Double-check your work and use online resources to verify your answers
• Not checking the conditions for the existence of an inverse function
  • Tip: Use the horizontal line test to ensure the function is one-to-one
• Overcomplicating the problem by using unnecessary techniques
  • Tip: Identify the most efficient method for the given problem

Practice Problems

1. Find the inverse of the function $f(x) = \frac{3x - 2}{x + 1}$
2. Determine the derivative of $y = \log_5(2x + 1)$
3. Use logarithmic differentiation to find $\frac{d}{dx} (x^2 + 1)^3$
4. Calculate the derivative of $y = \arccos(3x - 1)$
5. If $f(x) = e^{2x}$ and $g(x) = \ln(x + 1)$, find $(f \circ g)'(x)$
6. Evaluate $\lim_{x \to 0} \frac{\ln(1 + 2x)}{x}$
7. Prove that $\frac{d}{dx} \arctan(x) = \frac{1}{1 + x^2}$
8. Find the equation of the tangent line to the curve $y = xe^{-x}$ at the point $(1, \frac{1}{e})$
9. Determine the value of $x$ for which the function $y = x\log_2(x)$ has a horizontal tangent
10. Solve the equation $e^{2x} - 3e^x + 2 = 0$ using the substitution $u = e^x$