Grade 11

Derivative

The derivative measures how fast a quantity is changing at a single instant, built from the limit of average rates of change over shrinking intervals. It gives the slope of a tangent line, the instantaneous velocity of a moving object, and — through a few algebraic rules — a way to differentiate almost any function built from elementary pieces.

IntuitionFrom average speed to instantaneous speed

A car's speedometer reports how fast it is going right now, not its average speed over the whole trip. If a quantity is described by a function $y=f(x)$ , the average rate of change between $x_0$ and $x_0+h$ is the slope of the straight line (the secant line) joining the two points $(x_0, f(x_0))$ and $(x_0+h, f(x_0+h))$ : $\dfrac{f(x_0+h)-f(x_0)}{h}$ . This ratio depends on how large the step $h$ is. To capture the rate of change at the single instant $x_0$ , shrink the step: let $h \to 0$ and watch the secant line pivot around $(x_0, f(x_0))$ until it settles into the tangent line — the best straight-line approximation to the curve at that point.

The secant line through

(x_0, f(x_0))

and

(x_0+h, f(x_0+h))

for a shrinking step

h

. Drag

h

toward

0

to watch the secant pivot into the tangent line at

x_0

; its slope is

\dfrac{f(x_0+h)-f(x_0)}{h} \to f'(x_0)

SchoolThe limit definition of the derivative

Definition: Derivative at a point

Let $f$ be a function defined on an open interval containing $x_0$ . The **derivative of $f$ at $x_0$ **, written $f'(x_0)$ , is the limit of the difference quotient as the step shrinks to zero: $f'(x_0) = \lim_{h \to 0} \dfrac{f(x_0+h)-f(x_0)}{h}$ , equivalently $f'(x_0) = \lim_{x \to x_0} \dfrac{f(x)-f(x_0)}{x-x_0}$ . When this limit exists (and is finite), $f$ is called differentiable at $x_0$ . Other common notations for the derivative include $y'$ , $\dfrac{dy}{dx}$ , and $\dfrac{df}{dx}(x_0)$ .

f'(x_0) = \lim_{h \to 0} \frac{f(x_0+h)-f(x_0)}{h} = \lim_{x \to x_0} \frac{f(x)-f(x_0)}{x-x_0}

Geometric meaning. Since the tangent line at $x_0$ is the limit of secant lines, its slope is exactly $f'(x_0)$ . The tangent line therefore has equation $y = f(x_0) + f'(x_0)(x-x_0)$ — the best linear approximation to the graph of $f$ near $x_0$ . A horizontal tangent ( $f'(x_0)=0$ ) marks a point where the curve is momentarily flat, such as at a local extremum.

y = f(x_0) + f'(x_0)(x - x_0)

Physical meaning. If $s(t)$ is the position of an object at time $t$ , the average rate of change $\dfrac{s(t_0+h)-s(t_0)}{h}$ is average velocity over $[t_0, t_0+h]$ , and its limit $v(t_0)=s'(t_0)$ is the instantaneous velocity at $t_0$ . Differentiating again gives the acceleration $a(t) = v'(t) = s''(t)$ , the second derivative of position — the rate at which velocity itself is changing.

Theorem: Differentiability implies continuity

If $f$ is differentiable at $x_0$ , then $f$ is continuous at $x_0$ .

Why is it true?

A curve with a well-defined tangent slope at $x_0$ cannot jump there: write $f(x)-f(x_0) = \dfrac{f(x)-f(x_0)}{x-x_0} \cdot (x-x_0)$ ; as $x \to x_0$ the first factor tends to the finite number $f'(x_0)$ while the second tends to $0$ , so the product tends to $0$ and $f(x) \to f(x_0)$ .

Proof

This is exactly the argument above: the limit of a product is the product of the limits when both exist, and $\lim_{x\to x_0}\frac{f(x)-f(x_0)}{x-x_0}=f'(x_0)$ is finite by hypothesis while $\lim_{x\to x_0}(x-x_0)=0$ .

The converse is false: continuity does not imply differentiability. The standard counterexample is $f(x) = |x|$ at $x_0 = 0$ . It is continuous there, but the difference quotient $\dfrac{f(0+h)-f(0)}{h} = \dfrac{|h|}{h}$ equals $1$ for $h>0$ and $-1$ for $h<0$ , so the left- and right-hand limits disagree and $f'(0)$ does not exist — the graph has a sharp corner, with no single well-defined tangent slope.

UndergraduateDifferentiation rules

Basic differentiation rules
Rule	Formula	Condition / note
Constant multiple	$(cf)' = c f'$	$c$ is a constant
Sum / difference	$(f \pm g)' = f' \pm g'$	$f, g$ differentiable
Power rule	$(x^n)' = n x^{n-1}$	$n \in \mathbb{R}$ , $x$ in the domain
Product rule	$(fg)' = f'g + fg'$	$f, g$ differentiable at $x$
Quotient rule	$\left(\dfrac{f}{g}\right)' = \dfrac{f'g - fg'}{g^2}$	$g(x) \neq 0$
Chain rule	$(f \circ g)'(x) = f'(g(x))\, g'(x)$	$g$ differentiable at $x$ , $f$ differentiable at $g(x)$

Theorem: Product rule

If $f, g$ are differentiable at $x$ , then $fg$ is differentiable at $x$ and $(fg)'(x) = f'(x)g(x) + f(x)g'(x)$ .

Why is it true?

Picture $f(x)g(x)$ as the area of a rectangle with width $f(x)$ and height $g(x)$ . As $x$ moves, both side lengths change at once, so the rectangle's area changes for two reasons simultaneously: the width grows at rate $f'(x)$ while the height stays at $g(x)$ , contributing $f'(x)g(x)$ ; and the height grows at rate $g'(x)$ while the width stays at $f(x)$ , contributing $f(x)g'(x)$ . A common mistake is to guess $(fg)' = f'g'$ , but that ignores these two separate contributions entirely.

Proof

Write $\dfrac{f(x+h)g(x+h)-f(x)g(x)}{h}$ and insert $-f(x+h)g(x)+f(x+h)g(x)$ in the numerator to split it into $f(x+h)\cdot\dfrac{g(x+h)-g(x)}{h} + \dfrac{f(x+h)-f(x)}{h}\cdot g(x)$ . As $h \to 0$ , the first difference quotient tends to $g'(x)$ and $f(x+h) \to f(x)$ by continuity, while the second tends to $f'(x)$ ; the sum tends to $f(x)g'(x) + f'(x)g(x)$ .

Theorem: Chain rule

If $g$ is differentiable at $x$ and $f$ is differentiable at $g(x)$ , then $f \circ g$ is differentiable at $x$ and $(f \circ g)'(x) = f'(g(x))\, g'(x)$ .

Why is it true?

Rates of change compose by multiplying: if $y=f(u)$ changes twice as fast as $u$ (that is $f'=2$ ), and $u=g(x)$ changes three times as fast as $x$ (that is $g'=3$ ), then over a small step in $x$ , $u$ moves $3$ times as far, and $y$ then moves $2$ times as far as $u$ moved — so $y$ moves $2 \times 3 = 6$ times as far as $x$ did. A very common mistake is to forget the inner derivative $g'(x)$ : $\dfrac{d}{dx}\sin(x^2) \neq \cos(x^2)$ ; the correct derivative is $2x\cos(x^2)$ .

Proof

Since $f$ is differentiable at $g(x)$ , write $f(g(x)+k) - f(g(x)) = [f'(g(x)) + \varepsilon(k)]\,k$ for $k$ near $0$ , where $\varepsilon(k) \to 0$ as $k \to 0$ . Substituting $k = g(x+h)-g(x)$ , which tends to $0$ as $h \to 0$ by continuity of $g$ , and dividing by $h$ gives $\dfrac{f(g(x+h))-f(g(x))}{h} = [f'(g(x))+\varepsilon(k)]\cdot\dfrac{g(x+h)-g(x)}{h} \to f'(g(x))\,g'(x)$ .

y = f(x) = x^3 - 3x

, with

f'(x) = 3x^2 - 3

. The marked local maximum and minimum sit exactly where

f'(x)=0

, i.e.

x=\pm 1

: the curve rises while

f'(x)>0

(outside

[-1,1]

) and falls while

f'(x)<0

(inside

[-1,1]

). The marked inflection point at

x=0

is where

f''(x)=6x

changes sign, i.e. where the curve switches from concave down to concave up.

Example: Differentiating a product

Find $f'(x)$ for $f(x) = x^2 \sin x$ .

Solution

Apply the product rule with $u(x) = x^2$ and $v(x) = \sin x$ : $u'(x) = 2x$ and $v'(x) = \cos x$ , so $f'(x) = u'(x)v(x) + u(x)v'(x) = 2x\sin x + x^2\cos x$ .

Example

A ball is thrown upward from ground level. Its height in metres is $h(t) = 20t - 5t^2$ . (a) Find the instantaneous velocity $v(t) = h'(t)$ . (b) When does the ball reach its maximum height? (c) What is the speed at $t = 1$ s?

Solution

**Step 1 — Differentiate $h(t) = 20t - 5t^2$ using the power rule.** Each term differentiates separately: $(20t)' = 20$ and $(5t^2)' = 10t$ , giving $v(t) = h'(t) = 20 - 10t$ .

Step 2 — Find the time of maximum height. At the peak the ball is momentarily at rest, so $v(t) = 0$ : $20 - 10t = 0 \Rightarrow t = 2$ s. The second derivative $h''(t) = -10 < 0$ confirms this is a maximum.

**Step 3 — Evaluate the speed at $t = 1$ s.** Substituting into the velocity formula: $v(1) = 20 - 10(1) = 10$ m/s upward. The ball is still climbing at $t = 1$ s, consistent with reaching the peak at $t = 2$ s.

Historical note

In the late 17th century, Isaac Newton and Gottfried Wilhelm Leibniz independently developed the calculus of derivatives, but described it in strikingly different language: Newton spoke of fluxions $\dot{x}$ , the instantaneous rate at which a changing quantity flows, while Leibniz introduced the notation $\dfrac{dy}{dx}$ as a ratio of infinitesimal differences — the notation that survives today. Their followers argued for decades over priority and over what an "infinitely small" quantity even meant. It took over a century to settle: in 1823, Augustin-Louis Cauchy replaced infinitesimals with the rigorous $\varepsilon$ – $\delta$ limit definition used above, finally placing the derivative on solid logical footing.

Isaac Newton Gottfried Wilhelm Leibniz Augustin-Louis Cauchy

AdvancedHigher-order derivatives and where this leads

Everything above rests on the limit of a function, so a solid grasp of limits is the prerequisite for differentiation. Differentiating $f'$ itself gives the second derivative $f''=(f')'$ (acceleration, curvature), and repeating the process gives $f^{(n)}$ , the $n$ -th derivative; a function with continuous derivatives up to order $k$ is called $C^k$ , and $C^\infty$ if it has derivatives of every order. Two directions open up from here: reading off a function's shape — where it increases, has extrema, or is concave — directly from the sign of $f'$ and $f''$ (see applications of the derivative), and running the process in reverse: given $f'$ , recovering $f$ is the problem of antiderivatives, the gateway to integral calculus.

ResearchDerivatives at the research frontier

Research frontier as of 2026

As of 2026, three active directions extend differentiation far beyond hand-computed derivatives. (1) Automatic differentiation (AD). Modern machine learning trains neural networks with millions or billions of parameters by mechanically applying the chain rule to the sequence of elementary operations a program performs; reverse-mode AD, known in this setting as backpropagation, computes the exact gradient of the network's loss function in roughly the same time as one forward pass, and is implemented in differentiable-programming frameworks such as PyTorch and JAX (see Baydin, Pearlmutter, Radul & Siskind's 2018 survey). (2) Differentiating through non-smooth functions. Components like $\mathrm{ReLU}(x)=\max(x,0)$ are not differentiable everywhere, yet AD is routinely applied to them; Clarke's generalized gradient (1975) and recent work of Bolte and Pauwels (2020) give a rigorous mathematical foundation — a conservative field — explaining when and why automatic differentiation still produces mathematically meaningful, usable directions for such non-smooth programs. (3) Fractional derivatives. Riemann–Liouville and Caputo operators generalize $f^{(n)}$ to a derivative of arbitrary real (or complex) order $\alpha$ , an active tool for modelling anomalous diffusion and viscoelastic materials whose memory effects ordinary integer-order derivatives cannot capture.

Using the limit definition, what is $f'(3)$ for $f(x) = x^2$ ?

Why is $f(x) = |x|$ continuous but not differentiable at $x=0$ ?

By the chain rule, what is the derivative of $y = \sin(2x^2)$ ?

A particle moves along a line with position $s(t) = t^3 - 6t^2 + 9t$ (meters), $t \ge 0$ in seconds. At what time is its acceleration equal to zero?

References

Michael Spivak (2008). Calculus
James Stewart (2015). Calculus: Early Transcendentals
Augustin-Louis Cauchy (1823). Résumé des leçons données à l'École royale polytechnique sur le calcul infinitésimal
Atilim Gunes Baydin, Barak A. Pearlmutter, Alexey Andreyevich Radul, Jeffrey Mark Siskind (2018). Automatic Differentiation in Machine Learning: a Survey