From First Principles — watch the constant, power, constant-multiple, and sum rules each fall straight out of the limit definition.
We can always find a derivative from the limit definition, but doing that for every function is slow. Instead we'll run the definition once on each kind of function and watch a reusable rule drop out. By the end you'll be able to differentiate any polynomial on sight — and you'll know exactly why each shortcut works.
1. The Limit Definition
Every rule below starts here. The derivative is the limit of the difference quotient — the slope of the secant through $(x, f(x))$ and $(x+h, f(x+h))$ as the gap $h$ shrinks to zero.
Definition. The derivative of $f$ at $x$ is
the slope of the tangent line — the instantaneous rate of change.
As $h \to 0$ the second point slides into the first, and the secant becomes the tangent. Its slope is $f'(x)$. Now we feed specific functions into this machine.
2. The Constant Rule
Take $f(x) = c$, a flat horizontal line. Feed it into the definition: the numerator is $f(x+h) - f(x) = c - c$, which is just $0$.
The Constant Rule
Remember: flat lines don't change — a horizontal line has zero slope everywhere.
3. The Power Rule
The big one: the derivative of $x^n$. Expand $(x+h)^n$ with the binomial theorem and watch the pieces sort themselves — only one survives the limit.
The leading $x^n$ cancels with the $-x^n$; divide every remaining term by $h$; then let $h \to 0$ and every term that still carries an $h$ vanishes.
The Power Rule
Remember: pull the power down out front, then drop the exponent by one.
It works for any exponent:
4. The Constant Multiple Rule
Scaling a function by a constant $k$ scales its difference quotient by the same $k$ — so the constant just slides out in front of the limit, untouched.
The Constant Multiple Rule
Remember: constants are passengers — they ride along but don't change the differentiation.
Example
5. The Sum and Difference Rules
For $f(x) + g(x)$, the single difference quotient splits cleanly into one quotient for $f$ plus one for $g$. The limit of a sum is the sum of the limits, so the derivatives simply add.
The Sum / Difference Rules
Remember: differentiate term by term, keeping each $+$ or $-$.
All four together. Constant, power, constant-multiple, and sum are everything you need to differentiate any polynomial:
✓︎
Lesson complete
You've watched all four basic derivative rules fall out of one limit definition — and you can now differentiate any polynomial by inspection.
The derivative is the limit of the difference quotient — the secant becoming the tangent as $h \to 0$.
Constant rule: $\frac{d}{dx}[c] = 0$ — a flat line has zero slope.
Power rule: $\frac{d}{dx}[x^n] = n x^{n-1}$ — only the binomial's $h^1$ term survives.
Constant multiple: $\frac{d}{dx}[k\,f] = k\,f'$ — the constant slides outside the limit.
Sum / difference: $\frac{d}{dx}[f \pm g] = f' \pm g'$ — differentiate term by term.