∬Differential Calculus Unit 5 – Differentiation Rules and Change Rates

Key Concepts and Definitions

• Derivative represents the instantaneous rate of change of a function with respect to its input variable
• Differentiation is the process of finding the derivative of a function
• Constant Rule states that the derivative of a constant is always zero
• Power Rule allows for differentiating functions of the form $x^n$ by multiplying the exponent by the coefficient and then reducing the exponent by one
• Chain Rule is used to differentiate composite functions by multiplying the outer function's derivative by the inner function's derivative
• Leibniz Notation denotes the derivative of $f(x)$ with respect to $x$ as $\frac{d}{dx}f(x)$
• Higher-order derivatives are obtained by differentiating the function multiple times (first derivative, second derivative, etc.)
• Implicit differentiation is a technique used to find the derivative of a function that is not explicitly defined as a function of the independent variable

Basic Differentiation Rules

• Sum Rule states that the derivative of a sum of functions is equal to the sum of their individual derivatives
  • Example: If $f(x) = x^2 + \sin(x)$, then $f'(x) = 2x + \cos(x)$
• Difference Rule states that the derivative of a difference of functions is equal to the difference of their individual derivatives
• Product Rule is used to differentiate the product of two functions, $u(x)$ and $v(x)$, as $\frac{d}{dx}(u(x)v(x)) = u'(x)v(x) + u(x)v'(x)$
  • Example: If $f(x) = (x^2 + 1)(x - 2)$, then $f'(x) = (2x)(x - 2) + (x^2 + 1)(1)$
• Quotient Rule is used to differentiate the quotient of two functions, $\frac{u(x)}{v(x)}$, as $\frac{v(x)u'(x) - u(x)v'(x)}{[v(x)]^2}$
• Constant Multiple Rule states that the derivative of a constant multiple of a function is equal to the constant multiplied by the derivative of the function
• Power Rule for functions of the form $x^n$ states that the derivative is $nx^{n-1}$
  • Example: If $f(x) = x^5$, then $f'(x) = 5x^4$

Advanced Differentiation Techniques

• Chain Rule is used to differentiate composite functions by multiplying the outer function's derivative by the inner function's derivative
  • Example: If $f(x) = (2x + 1)^3$, then $f'(x) = 3(2x + 1)^2 \cdot 2$
• Implicit Differentiation is used when a function is not explicitly defined as a function of the independent variable
  • Differentiate both sides of the equation with respect to the independent variable
  • Solve the resulting equation for the derivative of the dependent variable
• Logarithmic Differentiation is useful for differentiating functions that involve products, quotients, or powers of functions
  • Take the natural logarithm of both sides of the equation
  • Use properties of logarithms to simplify the equation
  • Differentiate both sides of the equation using the Chain Rule
  • Solve for the derivative of the original function
• Parametric Differentiation is used when a curve is defined by parametric equations $x(t)$ and $y(t)$
  • Find $\frac{dx}{dt}$ and $\frac{dy}{dt}$ separately
  • Use the quotient rule to find $\frac{dy}{dx} = \frac{\frac{dy}{dt}}{\frac{dx}{dt}}$
• Higher-order derivatives are obtained by differentiating the function multiple times
  • Example: If $f(x) = x^3$, then $f'(x) = 3x^2$, $f''(x) = 6x$, and $f'''(x) = 6$

Applications to Rate of Change

• Velocity is the rate of change of position with respect to time, represented by the derivative of the position function
• Acceleration is the rate of change of velocity with respect to time, represented by the derivative of the velocity function (or the second derivative of the position function)
• Marginal cost is the rate of change of the total cost with respect to the quantity produced, represented by the derivative of the total cost function
• Marginal revenue is the rate of change of the total revenue with respect to the quantity sold, represented by the derivative of the total revenue function
• Population growth rate is the rate of change of the population with respect to time, represented by the derivative of the population function
• Optimization problems involve finding the maximum or minimum value of a function, which often requires setting the derivative equal to zero and solving for the input variable
  • Example: Minimizing the cost of materials for a cylindrical can with a fixed volume

Graphical Interpretation

• The derivative of a function at a point represents the slope of the tangent line to the function's graph at that point
• A positive derivative indicates an increasing function, while a negative derivative indicates a decreasing function
• A derivative of zero indicates a horizontal tangent line and a possible local maximum, local minimum, or inflection point
• Concavity of a function's graph is determined by the sign of the second derivative
  • Positive second derivative indicates concave up
  • Negative second derivative indicates concave down
• Inflection points occur where the concavity changes, corresponding to points where the second derivative is zero or undefined
• Sketching the graph of a function using information from its first and second derivatives (increasing/decreasing intervals, local extrema, concavity, and inflection points)
• Visualizing the relationship between a function and its derivative (derivative graph is above the x-axis when the original function is increasing, below the x-axis when decreasing)

Common Mistakes and How to Avoid Them

• Forgetting to apply the Chain Rule when differentiating composite functions
  • Identify the outer and inner functions and differentiate accordingly
• Misapplying the Product or Quotient Rule by differentiating each function separately and then multiplying or dividing
  • Follow the proper formulas for the Product and Quotient Rules
• Incorrectly differentiating negative exponents using the Power Rule
  • Rewrite the function with a positive exponent in the denominator before applying the Power Rule
• Confusing the derivative of $e^x$ (which is $e^x$) with the derivative of $a^x$ (which is $a^x \ln(a)$)
  • Remember that $e$ is a special case due to its unique properties
• Misinterpreting the signs of the first and second derivatives when analyzing the behavior of a function
  • A positive first derivative indicates an increasing function, while a negative second derivative indicates concave down
• Failing to consider the domain of a function when differentiating
  • Be aware of any restrictions on the input variable that may affect the differentiability of the function

Practice Problems and Solutions

1. Find the derivative of $f(x) = (3x^2 - 2x + 1)^4$
   • Solution: $f'(x) = 4(3x^2 - 2x + 1)^3 \cdot (6x - 2)$
2. Find the derivative of $g(x) = \frac{x^2 + 1}{\sqrt{x}}$
   • Solution: $g'(x) = \frac{(2x)(\sqrt{x}) - (x^2 + 1)(\frac{1}{2\sqrt{x}})}{x}$
3. Find the equation of the tangent line to the curve $y = \sin(2x)$ at $x = \frac{\pi}{6}$
   • Solution: The slope of the tangent line is $\cos(2 \cdot \frac{\pi}{6}) = \frac{\sqrt{3}}{2}$, and the point of tangency is $(\frac{\pi}{6}, \sin(\frac{\pi}{3})) = (\frac{\pi}{6}, \frac{\sqrt{3}}{2})$. The equation of the tangent line is $y - \frac{\sqrt{3}}{2} = \frac{\sqrt{3}}{2}(x - \frac{\pi}{6})$
4. Find the absolute maximum and minimum values of $f(x) = x^3 - 3x^2 - 9x + 5$ on the interval $[-2, 4]$
   • Solution: $f'(x) = 3x^2 - 6x - 9$. Critical points occur at $x = -1$ and $x = 3$. Evaluate $f(x)$ at the critical points and the endpoints of the interval: $f(-2) = -37$, $f(-1) = 12$, $f(3) = -32$, $f(4) = -27$. The absolute maximum is 12 at $x = -1$, and the absolute minimum is -37 at $x = -2$

Real-World Applications

• Optimization in business and economics
  • Maximizing profit by finding the optimal production quantity or pricing strategy
  • Minimizing costs by determining the most efficient allocation of resources
• Physical sciences and engineering
  • Analyzing the motion of objects using position, velocity, and acceleration functions
  • Determining the rates of change in chemical reactions or heat transfer processes
• Biological and environmental sciences
  • Modeling population growth and decay using exponential functions and their derivatives
  • Studying the rates of change in ecological systems, such as nutrient cycling or predator-prey interactions
• Social sciences and finance
  • Examining the rates of change in economic indicators, such as GDP or inflation
  • Analyzing the growth of investments using compound interest formulas and their derivatives
• Medicine and pharmacology
  • Investigating the rates of drug absorption and elimination in the body
  • Modeling the spread of diseases using differential equations and their solutions