Calculus Lecture: Derivatives of Functions

Basics of Derivatives

Constant Functions

• Derivative of any constant: Equals 0
  • Example: Derivative of 5 is 0, derivative of -7 is 0
• Graph interpretation: A constant function, like f(x) = 8, is a horizontal line. The slope of this line is 0.
  • f'(x) = 0 for any constant function
  • D/dx denotes differentiation with respect to x

Definition of a Derivative

• A derivative gives the slope at any given x value
• Notation: f'(x) or d/dx [f(x)] means the derivative of f(x)

Power Rule for Monomials

• Power Rule Formula: d/dx [x^n] = n * x^(n-1)
  • Example: d/dx [x^2] = 2x
  • Apply this rule to find derivatives of x^3, x^4, and x^5:
    • d/dx [x^3] = 3x^2
    • d/dx [x^4] = 4x^3
    • d/dx [x^5] = 5x^4*

Constant Multiple Rule

• Rule: Derivative of a constant multiplied by a function is the constant multiplied by the derivative of the function
  • Example: d/dx [4x^7] = 4 * d/dx [x^7] = 28x^6
  • Additional examples:
    • d/dx [8x^4] = 32x^3
    • d/dx [5x^6] = 30x^5
    • d/dx [9x^5] = 45x^4
    • d/dx [6x^7] = 42x^6*

Advanced Derivative Techniques

Limits and Derivatives

• Definition of the Derivative Using Limits: f'(x) = lim(h→0) [(f(x+h) - f(x))/h]
  • Example: Show d/dx [x^2] using limits
    • f(x) = x^2
    • f(x+h) = (x+h)^2 = x^2 + 2xh + h^2
    • Plug into limit definition and simplify to get 2x

Tangent and Secant Lines

• Tangent Line: Touches the curve at a single point, slope given by the derivative at that point
  • Example: Slope of y=x^2 at x=1 is 2
• Secant Line: Touches the curve at two points, approximate slope using average values
  • Example: Slope between (0.9, 0.81) and (1.1, 1.21) for y=x^2 is approximately 2
  • Can use closer points for better approximation

Polynomial Functions

• Finding the derivative of each term separately
  • Example: f(x) = x^3 + 7x^2 - 8x + 6, f'(x) = 3x^2 + 14x - 8
  • Another example: f(x) = 4x^5 + 3x^4 + 9x - 7, f'(x) = 20x^4 + 12x^3 + 9

Rational Functions

• Rewriting the function before differentiating
  • Example: f(x) = 1/x, rewrite as x^(-1), then differentiate
  • f'(x) = -1/x^2
  • Practice examples: 1/x^2, 8/x^4
    • Rewrite: x^(-2), 8x^(-4)
    • Apply power rule

Radical Functions

• Rewriting radical as rational exponents
  • Example: √x = x^(1/2), d/dx [√x] = (1/2)x^(-1/2) = 1/(2√x)
  • Practice: d/dx [∛(x^5)] = 5/3 * x^(2/3)*

Product and Quotient Rules

Product Rule

• Formula: d/dx [u*v] = u'v + uv'
• Example: d/dx [x^2 * sin(x)] = 2xsin(x) + x^2cos(x)
  • Multiple products: Differentiate terms one at a time

Quotient Rule

• Formula: d/dx [u/v] = (vu' - uv')/v^2
• Example: d/dx [(5x+6)/(3x-7)]
  • f'(x) = [(3x-7)*5 - (5x+6)*3]/(3x-7)^2
  • Simplify to get final answer

Derivatives of Trigonometric Functions

• Important rules to remember:
  • d/dx [sin(x)] = cos(x)
  • d/dx [cos(x)] = -sin(x)
  • d/dx [tan(x)] = sec^2(x)
  • d/dx [cot(x)] = -csc^2(x)
  • d/dx [sec(x)] = sec(x)tan(x)
  • d/dx [csc(x)] = -csc(x)cot(x)