The derivative as a function
What you'll learn
Read f′(x) as its own function: positive where f rises, negative where it falls, zero at the turns.
The derivative f′(x) isn't a single number — it's a new function. Feed it any x and it returns the slope of f there. Seeing f′ as a function of its own is the shift that makes the rest of calculus click.
f and f′, side by side
Read the two graphs together and the link is exact:
- Where f is rising, its slope is positive, so f′ sits above its axis.
- Where f is falling, its slope is negative, so f′ sits below.
- At each turning point of f, the slope is momentarily zero, so f′ crosses zero.
The bottom curve is just the top curve's slope, plotted point by point.
Notation
The same idea wears several notations — get comfortable with all of them:
| Notation | Read as | Style |
|---|---|---|
| f′(x) | "f prime of x" | Lagrange |
| dy/dx | "dee y dee x" | Leibniz |
| d/dx [ f(x) ] | "d by dx of f" | Leibniz (operator) |
Leibniz's dy/dx deliberately echoes the rise-over-run "Δy/Δx" of a slope — it remembers where the derivative came from.
Higher derivatives
Differentiate f′ and you get the second derivative, f″ — the rate at which the slope itself is changing. Position → velocity (f′) → acceleration (f″) is the classic chain. We'll put f″ to work on concavity soon.
Why this matters
Treating f′ as a function is what lets calculus answer questions about a whole curve at once: where it's steepest, where it levels off, where it turns. You stop computing one slope at a time and start reasoning about the slope everywhere.
Check your understanding
Question 1 of 2
Where f is decreasing, its derivative f′ is: