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What you will be able to do

1. 01State and apply the four elementary derivative facts $\frac{d}{dx}\sin x=\cos x$, $\frac{d}{dx}\cos x=-\sin x$, $\frac{d}{dx}e^x=e^x$, and $\frac{d}{dx}\ln x=\frac{1}{x}$ (for $x>0$), and combine them with the constant-multiple and sum rules.
2. 02Explain, from the shape of each graph, why these derivatives make sense: sine and cosine slopes read off the unit-circle curves, the exponential whose slope equals its own height, and the logarithm whose slope shrinks as $x$ grows.
3. 03Distinguish the exponential rule $\frac{d}{dx}e^x=e^x$ from the power rule, and avoid the two most common sign and structure errors — dropping the minus on $\frac{d}{dx}\cos x$ and treating $e^x$ as if the power rule applied.

Every derivative rule you have used until now — the power rule, the constant-multiple rule, the sum rule — was built for algebraic functions assembled from powers of $x$. The functions in this topic are different in kind. $\sin x$ and $\cos x$ do not settle down to a value; they oscillate forever. $e^x$ grows faster than any polynomial. $\ln x$ grows slower than any positive power. To do calculus on the real world — springs, populations, cooling coffee, radioactive decay — you need to differentiate these functions confidently, and that starts with four facts you will use for the rest of the course.