Derivatives of Exponential Functions

Exponential Functions

• Definition: An exponential function has the form y = a(c)^x, where (c) affects growth or decay.
  • Growth: (c > 1)
  • Decay: (0 < c < 1)
• Alternate form used: y = k \cdot a^x

Graph Characteristics

• Example Function: f(x) = 2^x
  • For (x < 0), function increases slowly; thus, f' values are small.
  • For (x > 0), function increases quickly; hence, f' values are larger.
• Always concave up and increasing for all (x).
• Expected Type of Derivative Function: Exponential

Tangent Line Calculation

• Method: Calculate slopes of tangent lines at selected points.
• Example Estimates (using graph):
  • At (x = -1), f'(-1) \approx 0.35
  • At (x = 0), f'(0) \approx 0.7
  • At (x = 1), f'(1) \approx 1.4
  • At (x = 2), f'(2) \approx 2.75
  • At (x = 3), f'(3) \approx 5.5

Numerical Estimation of Derivative

• Estimate f'(0):
  • Numerical limit: lim_{h \to 0} \frac{f(x + h) - f(x)}{h}
  • Calculation at (x = 0):
  • Derivative defined as:
    f'(0) = lim{h \to 0} \frac{2^{0+h} - 2^0}{h}= lim{h \to 0} \frac{2^h - 1}{h}
  • Results at small (h):
    • At (h = -0.0003): 0.6931
    • Approximate conclusion: f'(0) \approx 0.693

Patterns in Derivatives of Exponential Functions

• Observations:
  • For f(x) = 2^x, f'(x) \approx 0.693 \cdot 2^x
  • Generalization: For any base, f'(x) = f'(0) \cdot a^x.
• Sample Function: g(x) = 3^x
  • Based on calculations, g'(0) \approx 1.0986 \cdot 3^x

Derivative Relationships

• Functions whose derivative is proportional to themselves:
  • f(x) = a^x, f'(x) = k f(x).
  • Value of a is crucial; investigatory values lead to base e (approximately 2.718) as the only constant resulting in f'(x) = f(x).
  • Findings show that at zeros of their derivative function leads to:
    lim_{h \to 0}\frac{a^h - 1}{h} = k

Evaluating the Function y = e^x

• Features:
  • Zero x-intercepts
  • y-intercept: When (x = 0), y = 1.
  • Domain: {x | x \in \mathbb{R}}
  • Range: {y | y > 0, y \in \mathbb{R}}
  • Reflecting its inverse function: y = ln(x).

Logarithm Review

• Basic relationship: y = log_a(x)\Leftrightarrow a^y = x
• Natural Logarithm: ln(x), where base is e.
• ln and e are inverses.

Laws of Logarithms (base e)

• Product Law: ln(MN) = ln(M) + ln(N)
• Quotient Law: ln(\frac{M}{N}) = ln(M) - ln(N)
• Power Law: ln(M^p) = p \cdot ln(M)

Homework Problems

• Review exercises and examples provided in the transcript for practice.