Here's the most beautiful fact in this lesson, and you can watch it happen. Trace along \(\sin x\) and read off the slope at each point. Plot those slopes and they spell out a curve you already know — \(\cos x\). The derivative of sine literally is cosine.
What you're looking at: the lime curve is the function; the gold dot is its slope, dropping down to trace the derivative curve.
Watch where \(\sin\) is steepest (at \(x=0\)) — that's where \(\cos\) is highest. Where \(\sin\) peaks and flattens, \(\cos\) crosses zero. The slope story and the cosine curve are the same story.
Since \(\sin\) gives \(\cos\), what does \(\cos\) give? \(-\sin\). And that gives \(-\cos\), which gives \(\sin\) again. The derivatives of sine march around a four-step cycle forever. Step through it.
This is why higher-order derivatives of \(\sin\) and \(\cos\) are easy: just count steps around the cycle. The 100th derivative of \(\sin x\)? \(100 = 4\cdot25\), so you're back at \(\sin x\).
\(e^x\) is special in a way no other function is: its slope at every point equals its height at that point. Differentiating it changes nothing — \(\frac{d}{dx}e^x = e^x\). Drag along the curve and watch the height and the slope stay locked together.
What you're looking at: the blue bar is the height \(e^x\); the gold line is the tangent. Its slope always equals the bar's height.
They're identical at every \(x\) — the defining property of \(e^x\). Its inverse, \(\ln x\), has an equally tidy derivative, and a beautiful reason behind it — that's the very next section.
\(\tan\), \(\cot\), \(\sec\), \(\csc\) are all just combinations of \(\sin\) and \(\cos\). Their derivatives can be derived with the quotient rule — a tool you’ll learn next lesson — but for now, know the four results cold. Tap each to see it and a note on where it comes from.
Notice the symmetry: the three “co-” functions — \(\cos\), \(\cot\), \(\csc\) — all have negative derivatives. The other three are positive. That single pattern halves what you have to memorize.
The natural log is the inverse of \(e^x\) — reflect \(e^x\) across the line \(y=x\) and you get \(\ln x\). That reflection is the whole secret to its derivative. When you flip a curve across \(y=x\), you swap rise and run — so the slopes turn into reciprocals. That's why \(\frac{d}{dx}\ln x = \frac{1}{x}\).
What you're looking at: \(\ln x\) in lime and its mirror \(e^x\) in blue, reflected across the dashed \(y=x\). The two dots are reflections of each other.
Drag toward \(0\) and the slope shoots up — \(\ln x\) is nearly vertical there. Drag right and it flattens toward \(0\). The slope is always the reciprocal of the input: small \(x\), steep; large \(x\), gentle.
Each is the other’s derivative: \(\sin\to\cos\), \(\cos\to-\sin\). Cosine picks up the minus.
The one function where slope equals height everywhere. \(\frac{d}{dx}e^x=e^x\), forever unchanged.
The natural log’s derivative is the reciprocal: \(\frac{d}{dx}\ln x=\frac1x\). Short, clean, essential.
The three “co-” functions (cos, cot, csc) all have negative derivatives. The pattern halves the memorizing.
Five problems across the key functions — sine and cosine, \(e^x\) and \(\ln x\), a higher-order cycle, and the extra trig. Try each before revealing.
Eight fast checks across the whole lesson.
It's \(-\sin x\), not \(\sin x\). Cosine's derivative carries the minus sign — the single most common slip on this material.
\(e^x\) is not \(x^n\) — the variable is in the exponent. Its derivative is just \(e^x\), not \(xe^{x-1}\).
\(\frac{d}{dx}\tan x=\sec^2 x\), but \(\frac{d}{dx}\sec x=\sec x\tan x\). Different functions, easy to swap — learn them as a pair.
\(\cot\) and \(\csc\) both have negative derivatives (\(-\csc^2x\) and \(-\csc x\cot x\)). If your trig derivative is missing a minus, check whether it's a co- function.
The cycle is \(\sin\to\cos\to-\sin\to-\cos\), period 4. For the \(n\)th derivative, reduce \(n\) mod 4 — don't lose track of where the minus signs fall.
The derivative of \(\ln x\) is \(\frac1x\). It is not the log of anything, and it's not \(x\).