4.2 Definition of the derivative

The derivative is a key concept in calculus, measuring how a function changes at any point. It's defined as the limit of a difference quotient and represents the instantaneous rate of change or slope of the tangent line to a function's graph.

Calculating derivatives involves applying the limit definition to various functions. The derivative has practical applications in solving real-world problems, including rates of change and optimization. It's a powerful tool for analyzing function behavior and finding maximum or minimum values.

Definition and Interpretation of the Derivative

Definition of derivative using limits

• Defines the derivative of a function $f(x)$ at a point $x=a$ as the limit of the difference quotient as $h$ approaches 0:
  • $f'(a) = \lim_{h \to 0} \frac{f(a+h) - f(a)}{h}$
  • When the limit exists, it gives the instantaneous rate of change of $f(x)$ at $x=a$
• Derivative can be denoted using prime notation $f'(x)$, Leibniz notation $\frac{d}{dx}f(x)$, or $\frac{dy}{dx}$ when $y=f(x)$
• Finding the derivative is called differentiation

Interpretation of derivative as rate

• Derivative $f'(a)$ represents the instantaneous rate of change of $f(x)$ at $x=a$
  • Measures how quickly the function changes at that specific point ($x$-coordinate, time)
• Geometrically, $f'(a)$ is the slope of the tangent line to the graph of $f(x)$ at $(a, f(a))$
  • Tangent line touches the graph at a single point without crossing it (point of tangency)
• Derivative determines if a function is increasing ($f'(a) > 0$), decreasing ($f'(a) < 0$), or has a horizontal tangent ($f'(a) = 0$) at $x=a$

Calculating Derivatives and Applications

Calculation of derivatives for functions

• Steps to find the derivative using the limit definition:
  1. Write the definition of the derivative using the limit
  2. Substitute the given function into the definition
  3. Simplify the numerator by combining like terms or using function properties
  4. Factor out the common factor of $h$ in the numerator, if possible
  5. Evaluate the limit as $h$ approaches 0
• Examples of derivatives using the limit definition:
  • Polynomial function ($f(x) = x^2$): $f'(x) = \lim_{h \to 0} \frac{(x+h)^2 - x^2}{h} = \lim_{h \to 0} \frac{2xh + h^2}{h} = \lim_{h \to 0} (2x + h) = 2x$
  • Rational function ($f(x) = \frac{1}{x}$): $f'(x) = \lim_{h \to 0} \frac{\frac{1}{x+h} - \frac{1}{x}}{h} = \lim_{h \to 0} \frac{-h}{x(x+h)h} = \lim_{h \to 0} \frac{-1}{x(x+h)} = -\frac{1}{x^2}$
  • Trigonometric function ($f(x) = \sin(x)$): $f'(x) = \lim_{h \to 0} \frac{\sin(x+h) - \sin(x)}{h} = \lim_{h \to 0} \frac{2\cos(x+\frac{h}{2})\sin(\frac{h}{2})}{h} = \cos(x)$

Application of derivatives to problems

• Rates of change problems:
  • Derivative calculates the rate of change of one variable with respect to another (velocity, growth rate)
  • Example: Height of a balloon $h(t) = -16t^2 + 100t + 50$ (meters) at time $t$ (seconds), velocity at any time $t$ is $h'(t) = -32t + 100$ m/s
• Optimization problems:
  • Derivative finds maximum or minimum values of a function within a given domain
  • Steps to solve optimization problems:
    1. Identify the function to be optimized (profit, area, volume)
    2. Determine the domain based on given constraints
    3. Find the derivative of the function
    4. Set the derivative equal to zero and solve for critical points
    5. Evaluate the function at critical points and domain endpoints to find max/min value
  • Example: Minimize surface area $A(x, y) = 2(xy + \frac{1000}{xy} + \frac{1000y}{x})$ of a rectangular box with volume 1000 cubic inches, where $x$, $y$, $z$ are length, width, height, subject to $xyz = 1000$