Differentiation from first principles

What you are looking at

The derivative is defined as a limit, $f'(x) = \lim_{h \to 0} \dfrac{f(x+h)-f(x)}{h}$. The quotient is the gradient of the secant through $P = (x, f(x))$ and $Q = (x+h, f(x+h))$: the rise $f(x+h)-f(x)$ over the run $h$, both marked on the graph. You can drag $P$ along the curve, drag $Q$ to either side of it, or press Shrink $h$, which steps $h$ down through $0.5, 0.2, 0.1, \dots$ to $0.001$ while the secant swings onto the dashed tangent.

The working panel follows a written answer. It substitutes numbers into the quotient, then simplifies algebraically before letting $h \to 0$: for $x^2$, $\dfrac{(x+h)^2 - x^2}{h} = 2x + h \to 2x$; for $x^3 - 3x$ the quotient becomes $3x^2 + 3xh + h^2 - 3$; for $\sin x$ it rests on $\dfrac{\sin h}{h} \to 1$ and $\dfrac{\cos h - 1}{h} \to 0$. Set $h = 0$ and the secant disappears, because the quotient is undefined. Hide formulas turns the panel into a prediction question.

Try this

1. 1With $f(x) = x^2$ and $P$ at $x = 1$, press Shrink $h$: the gradient of $PQ$, which is $2 + h$, falls towards 2 as $h$ steps down to $0.001$.
2. 2Drag $Q$ to the left of $P$ so that $h$ is negative: the gradient now approaches 2 from below. The limit is the same from both sides.
3. 3Slide $h$ to exactly 0: the secant vanishes and the panel reports that the quotient is undefined, which is why the limit is taken only after simplifying.
4. 4Choose $\sin x$ and shrink $h$: $\dfrac{\sin h}{h}$ approaches 1 and $\dfrac{\cos h - 1}{h}$ approaches 0, leaving $f'(x) = \cos x$.

Mistakes students make

• Putting $h = 0$ straight into $\dfrac{f(x+h)-f(x)}{h}$. That gives $\dfrac{0}{0}$; expand, cancel the factor $h$ first, and only then let $h \to 0$.
• Expanding $(x+h)^2$ as $x^2 + h^2$. The correct expansion is $x^2 + 2xh + h^2$, and the $2xh$ term is the one that gives the derivative.
• Dropping the limit too early, or ending with $f'(x) = 2x + h$. Keep writing $\lim_{h \to 0}$ on every line until you actually let $h \to 0$.

Where it is taught

• IB Maths AA SL 5.1 · AA HL 5.12
• AP Calculus AB/BC Unit 2
• Edexcel A-Level Pure Year 1 · Differentiation from first principles
• Edexcel A-Level Pure Year 2 · Differentiating sin x and cos x from first principles
• Greek Lykeio year 3 (Γ΄ Λυκείου) · §2.1