AP Calculus: Derivatives of Exponential and Logarithmic Functions (e^x, ln x, a^x, log_a x)

Derivatives of Exponential and Logarithmic Functions (e^x, ln x, a^x, log_a x)

Concept Summary

• Derivative of e?: The derivative of e? is e?; this is the only function whose derivative is itself, making it fundamental in calculus.
• Derivative of ln x: The derivative of ln x is 1/x for x > 0; this arises from the inverse relationship with e?.
• Derivative of a?: The derivative of a? is a? ln a (where a > 0, a-1); generalizes the e? case using logarithmic differentiation.
• Derivative of log? x: The derivative of log? x is 1/(x ln a) for x > 0; derived via change of base or implicit differentiation.
• Logarithmic Differentiation: A technique to differentiate functions of the form f(x)^g(x) by taking the natural log of both sides before differentiating.

Core Questions

WHAT (definitional)

Q: What is the derivative of e?? A: The derivative of e? is e?. Trap/Clarification: Students often assume e?’s derivative is xe¹ (like x?), but e? is unique—its derivative is itself.

Q: What is the derivative of ln x? A: The derivative of ln x is 1/x for x > 0. Trap/Clarification: The domain restriction (x > 0) is critical; ln x is undefined for x-0, so its derivative doesn’t exist there.

WHY (causal/explanatory)

Q: Why is the derivative of e? equal to e?? A: The limit definition of e (lim_{h?0} (e?-1)/h = 1) ensures that the slope of e? at any point x is e?. Trap/Clarification: This is not true for other bases (a? requires ln a); e is the only base where the derivative is the function itself.

Q: Why is logarithmic differentiation useful? A: It simplifies differentiating products/quotients of functions or functions with variable exponents (e.g., x?) by converting them into sums/differences via ln properties. Trap/Clarification: Forgetting to exponentiate both sides at the end (to solve for dy/dx) is a common error.

HOW (process/application)

Q: How do you differentiate a?? A: Rewrite a? as e and use the chain rule: d/dx [a?] = a? ln a. Trap/Clarification: Misapplying the power rule (d/dx [a?] = xa¹) is incorrect; a? is an exponential, not a power function.

Q: How do you differentiate log? x? A: Use the change of base formula: d/dx [log? x] = 1/(x ln a). Trap/Clarification: Confusing log? x with ln x (which has derivative 1/x) leads to errors; the ln a in the denominator is essential.

Q: How do you use logarithmic differentiation for y = f(x)^g(x)? A: Take ln of both sides, differentiate implicitly, then solve for dy/dx: dy/dx = y · d/dx [ln f(x)^g(x)]. Trap/Clarification: Forgetting to multiply by y (the original function) at the end is a frequent mistake.

CAN (conditions/possibilities)

Q: Can the derivative of a? ever be negative? A: Yes, if 0 < a < 1 (since ln a < 0), the derivative a? ln a is negative for all x. Trap/Clarification: Assuming a? always has a positive derivative (like e?) ignores cases where a < 1.

Q: Under what conditions does d/dx [log? x] exist? A: The derivative exists for x > 0 and a > 0, a-1; log? x is undefined otherwise. Trap/Clarification: Students often overlook the a-1 restriction (since log? x is undefined).

Quick Facts & Traps

• Fact: d/dx [e] = ke (chain rule application).
• Trap: d/dx [x?] = ex¹-Reality: x? is a power function, not exponential; the derivative is ex¹.
• Fact: d/dx [ln |x|] = 1/x for x-0 (extends the domain to negative x).
• Trap: d/dx [ln(kx)] = 1/(kx)-Reality: d/dx [ln(kx)] = 1/x (the k cancels out).
• Fact: d/dx [a?] = a? ln a works for any a > 0, a-1.
• Trap: d/dx [log? x] = 1/x-Reality: The correct derivative is 1/(x ln a).

Rapid-Fire True/False

• Statement: The derivative of e? is e? only at x = 0. Answer: FALSE Why the common mistake happens: Confusing the limit definition (where e is defined via h?0) with the derivative’s behavior at a single point.

• Statement: d/dx [ln(x²)] = 1/x². Answer: FALSE Why the common mistake happens: Forgetting the chain rule; the correct derivative is 2/x.

• Statement: d/dx [2?] = x·2¹. Answer: FALSE Why the common mistake happens: Misapplying the power rule to an exponential function. The correct derivative is 2? ln 2.