Derivatives of cos x, sin x, e^x, and ln x AB & BC

Four results carry this topic: $\frac{d}{dx}[\sin x] = \cos x$, $\frac{d}{dx}[\cos x] = -\sin x$, $\frac{d}{dx}[e^x] = e^x$, and $\frac{d}{dx}[\ln x] = \frac{1}{x}$ for $x > 0$. The general versions attach a logarithm to the base: $\frac{d}{dx}[a^x] = a^x \ln a$, and $\frac{d}{dx}[\log_b x] = \frac{1}{x \ln b}$. Repeated differentiation of sine runs a four-step cycle, $\sin \to \cos \to -\sin \to -\cos$, and all of it assumes radians.

Two error families, and they are worth keeping apart. The minus sign in the trigonometric pair migrates onto the wrong member or drops out of a higher derivative, so $\frac{d}{dx}[\cos x]$ is reported as $\sin x$ or $\frac{d^2}{dx^2}[\sin x]$ as $\sin x$. Separately, the exponential and logarithmic family gets flattened: every base treated as if it were $e$, every logarithm differentiated as $\frac{1}{x}$ regardless of base, and the $\ln a$ factor placed in the wrong position.