Calculus AB · Differentiation: Definition and Fundamental Properties · 14 min read · Updated 2026-05-10
Derivatives of cos, sin, e^x, ln(x) — AP Calculus AB
AP Calculus AB · Differentiation: Definition and Fundamental Properties · 14 min read
1. Core Overview★★☆☆☆⏱ 3 min
This topic establishes four core derivative rules for the most common non-algebraic (transcendental) functions tested on AP Calculus AB. It makes up part of the 10–12% of the total exam score allocated to Unit 2, and appears in nearly every exam as both standalone multiple choice and a required intermediate step in longer free response questions.
You will use these rules to solve problems asking for derivatives of function combinations, tangent line slopes, and instantaneous rates of change. Unlike the power rule for algebraic functions, these rules are specific to each function and require memorization (rooted in first principles) to apply correctly. Mastery of these rules is non-negotiable for all later differentiation topics.
2. Derivatives of Sine and Cosine★★☆☆☆⏱ 4 min
Exam tip: Always double-check the sign on the derivative of cosine: the negative sign is the most commonly missed detail on trig derivative multiple choice questions. If your answer is off by a negative sign, this is almost always the error.
3. Derivative of the Natural Exponential $e^x$★★☆☆☆⏱ 3 min
Exam tip: Don't confuse the derivative of $e^x$ with the power rule: if you mistakenly apply the power rule to $e^x$, you'll get $xe^{x-1}$, which is incorrect.
4. Derivative of the Natural Logarithm $\ln x$★★★☆☆⏱ 4 min
Exam tip: Always remember the domain restriction for $\ln x$: if a question asks for the derivative of $\ln x$ at a non-positive $x$, the derivative does not exist, because $\ln x$ itself is undefined there. This is a common trick question on multiple choice.
5. Concept Check★★★☆☆⏱ 4 min
Common Pitfalls
Why: Students mix up the order of sine and cosine derivatives, memorizing both as positive.
Why: Students confuse the constant base $e$ with a variable base $x$ in power functions.
Why: Students mix up the derivative of $\ln x$ with the derivative of $e^x$ (which is itself), so they incorrectly assume $\ln x$ is also its own derivative.
Why: Students forget the domain of $\ln x$ is only positive $x$, so the function does not exist for non-positive inputs, so its derivative also does not exist there.
Why: Students see $\ln$ and automatically apply the derivative rule for $\ln x$, forgetting that $\ln 5$ is a constant number, not a function of $x$.