A derivative is itself a function — so you can differentiate it again. Do it once for
slope, twice for concavity and acceleration, a third time, and keep climbing. This lesson makes the whole tower of
derivatives something you can watch move at once.
Differentiating doesn’t have to stop at \(f'\). Since \(f'\) is a function in its own
right, differentiate it to get \(f''\) — the rate at which the slope itself is changing, which is exactly
concavity, and in motion problems, acceleration. Keep going for \(f'''\), \(f^{(4)}\), and beyond.
Each one is just the slope of the one above it.
Bend the curve, build the tower
This is the engine of the lesson. Reshape \(f\) with your hands, then build its tower of
derivatives one rung at a time — meet each one before the next appears. Hit differentiate to reveal the
slope \(f'\), then concavity \(f''\), then keep climbing.
f
Three ways to write it
The same idea wears three notations. You’ll see all of them on the AP exam, sometimes in a
single problem, so read them as synonyms.
Lagrange — primes
\(f'(x)\)
\(f''(x)\)
\(f'''(x),\ f^{(4)}(x),\ \dots\)
Leibniz — ratios
\(\dfrac{dy}{dx}\)
\(\dfrac{d^2y}{dx^2}\)
\(\dfrac{d^n y}{dx^n}\)
In motion
position \(s(t)\)
velocity \(s'(t)\)
acceleration \(s''(t)\)
Careful: \(\dfrac{d^2y}{dx^2}\) is not \(\left(\dfrac{dy}{dx}\right)^2\). The little \(2\)s mark
“differentiate twice,” not “square it.”
Take a ride: position, velocity, acceleration
Now the same tower, applied to motion. Below is the height of a roller-coaster car over time.
Press ride to send the car, then differentiate to reveal velocity (its speed) and once more for
acceleration — the push you feel into your seat. That is exactly where BC motion lives: position,
velocity, and acceleration.
s
Patterns that repeat (or never stop)
Some functions fall into a rhythm when you keep differentiating. Pick one and step it.
A worked tower
Find every derivative of \(s(t)=t^3-6t^2+9t\) (a particle’s position) and read off when it is
speeding up. Press through it.
press step
what each derivative tells you
\(f'\) — the slope; in motion, the velocity. Where it is zero, \(f\) has a flat tangent.1st
\(f''\) — the concavity; in motion, the acceleration. Its sign tells you which way \(f\) bends.2nd
\(f'''\) — the rate concavity changes. Differentiate once more and you keep climbing the tower.3rd
\(f^{(n)}\) — keep going. Polynomials run out; \(\sin\) and \(\cos\) cycle; \(e^x\) never changes.nth
The core ideas
Each is the slope of the one above
\(f''\) is just \(f'\) differentiated. Read the cascade top-down: every height below is the slope of the curve above it.
Second derivative = concavity
\(f''\gt 0\) means \(f\) curves upward (a valley shape); \(f''\lt 0\) curves downward. A sign change of \(f''\) is an inflection point.
Polynomials terminate
Each derivative drops the degree by one. A degree-\(n\) polynomial has \(f^{(n+1)}(x)=0\) everywhere — the tower bottoms out at zero.
Some never settle
\(\dfrac{d^n}{dx^n}e^x=e^x\) forever, and \(\sin,\cos\) cycle with period \(4\). These rhythms power Taylor series later.
Practice
Quick check 8 questions
Common traps
Confusing \(f''\) sign with \(f'\) sign
\(f'\) tells you increasing/decreasing; \(f''\) tells you concave up/down. A curve can be rising while concave down — slowing its climb.
Forgetting the chain rule compounds
For \(y=e^{2x}\), each derivative pulls another factor of \(2\): \(2e^{2x},\,4e^{2x},\,8e^{2x},\dots\) The constant grows, it doesn’t vanish.
Thinking polynomials never zero out
They do. \(\frac{d}{dx}\) lowers the degree each time, so a degree-\(3\) polynomial has \(f^{(4)}(x)=0\) everywhere.
Reading \(\frac{d^2y}{dx^2}\) as \(\left(\frac{dy}{dx}\right)^2\)
It means “differentiate twice,” not “square the first derivative.” The \(2\)s are operation counts.
Motion: velocity \(=s'\), acceleration \(=s''\). A particle speeds up when \(v\) and \(a\) share a sign.
For \(\sin\) and \(\cos\), reduce \(n\) mod \(4\); for \(e^{kx}\), the \(n\)th derivative is \(k^n e^{kx}\).
Watch the notation: \(f''(2)\) is a number (the second derivative at \(x=2\)), not a second point.
You can now climb the whole tower of derivatives — slope, concavity, acceleration, and the patterns behind series.
Next, the unit’s capstone: deciding which technique a messy derivative actually needs.
Selecting Procedures is next — or head back to the
Unit 3 Guide.