Derivatives of Exponential and Log Functions Exam Notes

True/False (2pts each)

• If y + xy = 2, then \frac{dy}{dx} = \frac{xy + y}{x}. This statement should be evaluated based on implicit differentiation.
• If y = x^4 e^x, then \frac{dy}{dx} = 4x^3 e^x. This statement requires the product rule.
• If y = h(x^2), then \frac{dy}{dx} = 77. This seems incomplete and lacks context about the function h(x^2).
• If y = \ln(\sin x), find \frac{dy}{dx}.

Multiple Choice (3pts each)

• If y = (\ln x)e^{3x}, then \frac{dy}{dx} = ?
  • a. \frac{e^{3x}}{x}
  • b. \frac{\cos x}{\sin x}
  • c. 3(\ln x)e^{3x}
  • d. 3(\ln x)e^{3x} + \frac{e^{3x}}{x}
• If y = \ln 8x, find \frac{dy}{dx}:
  • a. \frac{1}{x}
  • b. \frac{1}{8x}
  • c. \frac{8}{x}
  • d. \frac{8}{\ln 8x}
• Find the derivative of f(x) = \ln(\tan x):
  • a. f'(x) = \tan(\frac{x}{2})
  • b. f'(x) = \frac{\sec^2 x}{\tan x}
  • c. f'(x) = -\csc^2 x \cdot \cot x
  • d. f'(x) = \sec^2 x

Short Answer (4pts each)

• Find the derivative of y = 4 - 2x^2.
• Find the derivative of y = (4^{2x} + 2)^3.
• Find the derivative of f(x) = \log_2(4x^2).
• Find the derivative of y = (\ln x)^2.
• The spread of a rumor at Deep Run High School is modeled by the equation R(x) = \frac{2300}{1 + e^{-x}}, where R(x) is the number of people who know the rumor and x is the number of days the rumor has spread.
  • (a) Find R(0). What does this number mean?
  • (b) How fast is the rumor spreading after 3 days?

Practice Problems (7 points each)

• Find the derivative of f(x) = \log_3 \sqrt{\sqrt{1 + 6x}}.
• Find the derivative of y = \sqrt{x^3 + 1}(x^2 + 1)^4.
• Find the derivative of f(x) = \ln(4x^5).
• Find the derivative of f(x) = e^{3x^7}.
• Find the derivative of f(x) = 11^{(\sin 5x^3)}.
• Find the derivative of y = \sqrt{2x^3 - 7}.
• y = 3 \cot x
• y = x^3 e^{2x}
• y = \frac{\sqrt{x} - 5}{x^4 - 1}
• f(x) = \frac{e^x}{1 + x^2}
• f(x) = 4 + \ln x
• f(x) = \frac{1}{4} + \ln x
• y = \log_5(x^2 + 5x)
• y = \log_2 \sqrt{x}

Additional Problems

• Describe where the graph of 3x^2 + y^2 - 13x + 19y = 4 is horizontal and where it is vertical.
• If 3y - 2 = -12x^2 + 8\sin y, then find \frac{dy}{dx}.
• If 12x^2 + 14xy + 8y^2 = 12, then find \frac{dy}{dx}.
• Find \frac{d^2y}{dx^2} if x^2 + y^2 = 16.