Topic 2.7 Derivatives of cos x, sin x, e to the x, and ln x: state and apply the derivatives of the four basic transcendental functions.

What this topic is asking

The College Board (Topic 2.7) wants you to know the derivatives of the four core transcendental functions: $\sin x$, $\cos x$, $e^x$, and $\ln x$. These are memorized facts (derived from the limit definition and the special trig limits) that you combine with the linearity rules to differentiate a wide range of functions.

The four derivatives

The sine and cosine derivatives come from the limit definition together with the special limits $\lim_{x \to 0}\frac{\sin x}{x} = 1$ and $\lim_{x \to 0}\frac{1 - \cos x}{x} = 0$ from Unit 1. The exponential $e^x$ is the unique function equal to its own derivative, and $\ln x$ has the clean derivative $\frac{1}{x}$.

The signs to watch

Combining with linearity

Once you know the four derivatives, the constant-multiple and sum rules from Topic 2.6 let you differentiate any linear combination. The transcendental and power functions add together cleanly: each term is differentiated by its own rule, then summed.

Where these four derivatives come from

It is worth knowing the origin of each, because the AP exam sometimes probes understanding rather than recall. The sine and cosine derivatives fall out of the limit definition: expanding $\sin(x + h)$ with the angle-addition formula and rearranging produces terms multiplied by $\frac{\sin h}{h}$ and $\frac{1 - \cos h}{h}$, and the special Unit 1 limits ($1$ and $0$ respectively, in radians) collapse the whole expression to $\cos x$. The same route on cosine yields $-\sin x$, with the minus sign appearing naturally. The exponential $e^x$ is defined so that its rate of change equals its own value - that self-replicating property is what singles out the base $e$ from other bases - which is why $\frac{d}{dx}[e^x] = e^x$. The logarithm derivative $\frac{1}{x}$ is the inverse-function counterpart of the exponential. You do not need to reproduce these derivations on most questions, but understanding that the trig derivatives depend on radians explains why degrees are never used in calculus.

Where these lead

These four derivatives are the raw material for the product and quotient rules (Topics 2.8 and 2.9), where you differentiate things like $x^2 e^x$ or $\frac{\sin x}{x}$. They also underpin the derivatives of $\tan x$, $\sec x$ and the other trig functions (Topic 2.10), which are built from $\sin x$ and $\cos x$ via the quotient rule. A common exam pattern combines all of them at once: a single function such as $f(x) = e^x \sin x + 3\ln x$ tests whether you can pair the right derivative with the right factor and keep the cosine sign correct. Practicing mixed combinations until each derivative is automatic is the fastest way to stop losing easy marks on the no-calculator section, where these appear constantly as the inner pieces of larger problems.