Calculus BC · Unit 2: Differentiation: Definition and Fundamental Properties · 14 min read · Updated 2026-05-11
Derivatives of cos x, sin x, e^x, ln(x) — AP Calculus BC
AP Calculus BC · Unit 2: Differentiation: Definition and Fundamental Properties · 14 min read
1. Derivatives of Sine and Cosine★☆☆☆☆⏱ 4 min
The derivatives of $\sin x$ and $\cos x$ are derived directly from the limit definition of the derivative, using the two key trigonometric limits: $\lim_{h \to 0} \frac{\sin h}{h} = 1$ and $\lim_{h \to 0} \frac{\cos h - 1}{h} = 0$. These formulas only hold when $x$ is in radians, which is always the AP exam convention.
Exam tip: AP exam questions almost always require exact form for answers involving radicals and $\pi$, so never convert to a decimal unless explicitly asked.
2. Derivative of the Natural Exponential Function $e^x$★☆☆☆☆⏱ 3 min
The natural exponential function $f(x) = e^x$, where $e \approx 2.71828$ is Euler's number, has a unique property: it is its own derivative. This result holds for all real $x$, and only applies to the base $e$ natural exponential function.
3. Derivative of the Natural Logarithm $\ln(x)$★★☆☆☆⏱ 3 min
$\ln(x)$ is the inverse function of $e^x$, defined only for $x>0$. We use inverse function differentiation to derive its derivative.
Exam tip: Always check your simplification of tangent line equations: constant terms often cancel, and AP graders deduct points for incorrect final form even if the slope is correct.
4. Combined AP-Style Practice★★☆☆☆⏱ 4 min
Common Pitfalls
Why: Students confuse the order of trigonometric derivatives, mixing derivative rules with antiderivative rules.
Why: Students see an exponent and automatically use the power rule, which only works for constant exponents.
Why: Students mix up the derivative of $\ln x$ with other rules, or confuse it with exponential derivatives.
Why: Students apply the $\frac{1}{x}$ rule to any expression with $\ln$, forgetting that $\ln$ of a constant is just a constant.
Why: Students see the lack of a negative sign on $e^x$ or $\sin x$ and incorrectly carry the positive sign over to cosine.
Why: Students confuse the derivative of $\ln x$ with the power rule for $x^{-1}$, or misread the problem.