Approaching a value

A limit is a value approached by a function as its input approaches a specified value, even if the function doesn’t actually take that value there. We write

To find the limit, look at how the function behaves near $x=c$, rather than just at the plotted point when $x=c$. The graph might have a hole or a filled point somewhere else, yet the nearby values can still approach one common number. When the values from the left and right approach different numbers, the two-sided limit does not exist.

For a numerical estimate, draw up a table using inputs that get successively closer to $c$ from both sides. For a graphical estimate, follow the curve towards $c$ from the left, then from the right. Spreadsheets, graphing software and a GDC can help form and check a conjecture. Be careful with very large zoom factors, though, as they may reveal rounding or pixel limitations. Formal algebraic limit calculations are not required here.

From secants to a tangent

For $y=f(x)$, where $y$ is the output (measured in the relevant SI output unit), the gradient between two nearby points is

This gives the gradient of a secant line. As $h$ approaches zero, the second point moves towards the first. The secant gradients may then approach a limiting value.

A derivative is a function or value that gives the instantaneous rate at which one variable changes with respect to another. Informally,

The notation $f'(x)$ represents the derivative function. When $y=f(x)$, the same derivative can be written as $\frac{dy}{dx}$. Its unit is the unit of $y$ per unit of $x$. The limit definition explains the underlying idea; formal analytic manipulation of limits is not required.

Gradient and rate interpretations

Geometrically, $f'(x)$ gives the gradient of the tangent to the graph at input $x$. In context, it represents an instantaneous rate of change. Both its sign and unit matter. A rate of $-4\ \text{m s}^{-1}$ describes a decrease of four metres per second, not simply the number $-4$.

Match the notation to the variables used in the problem. For instance,

is the rate of change of volume with radius

Similarly,

is the rate of change of displacement with time

The units of these derivatives are therefore $\text{m}^2$ and $\text{m s}^{-1}$ respectively.

This interpretation also applies to marginal cost, revenue and profit in economics; motion, induced emf and oscillation models in physics; and gradients of experimental curves in chemistry.

Perspective

Calculus developed from attempts to reason about quantities becoming arbitrarily small. Centuries before European calculus, Indian mathematicians investigated zero and division by zero. Ancient Greek discomfort with zero, by contrast, limited the development of related geometric work. Newton and Leibniz later created different methods and notations, which led to a long priority dispute rather than a simple account of one isolated discoverer.

Limits turn intuitive ideas such as “instantaneous” and “arbitrarily close” into usable mathematics. Even so, the model remains an idealisation: no physical measurement observes an infinitely small interval. This raises a useful knowledge question—when does mathematically disciplined intuition count as justification, and when must it be replaced by proof or evidence?

Reading change from the derivative

An increasing function on an interval is a function whose output rises as the input moves from left to right throughout that interval. A decreasing function on an interval is a function whose output falls as the input moves from left to right throughout that interval.

The derivative shows the function’s local direction:

• $f'(x)>0$: the tangent slopes upwards, so $f$ is increasing;
• $f'(x)<0$: the tangent slopes downwards, so $f$ is decreasing;
• $f'(x)=0$: the tangent is horizontal at that input.

A derivative of zero only describes what is happening at one point, not across an entire interval. It also doesn’t prove that the function turns there. The derivative can be zero even when the function keeps increasing on both sides.

To find the intervals, use the derivative graph or solve for its zeros, then check its sign between them. Write the answers as intervals and respect the stated domain. On a GDC, graph $f'(x)$ and identify where it lies above or below the horizontal axis. Don’t confuse these regions with where the original graph $f(x)$ lies above or below its axis.

The integer power rule

For a power function,

Put simply: multiply by the exponent, then reduce the exponent by one.

This rule works for positive, zero and negative integer exponents. The derivative of a constant is zero. With negative powers, any original domain restrictions still apply. For example,

Polynomial and term-by-term differentiation

A polynomial is a finite sum of constant multiples of non-negative integer powers of a variable. The syllabus rule also applies term by term to more general expressions of the form

Addition and subtraction don’t change the process. Differentiate each term, keep its sign, then simplify.

Technology offers a useful check: graph the original function alongside the derivative you generated. The derivative should be positive where the original graph rises and negative where it falls. Technology can verify a derivative, but the power rule shows why the algebraic form makes sense.

Models and reality

Integer-power models can represent trajectories, changing costs and engineered systems accurately enough to support achievements such as spaceflight. Even so, the model isn’t identical to reality. It isolates selected variables and assumptions, and remains useful only while those choices are adequate over the domain in question.

Tangent equations

A tangent is a line whose gradient at a point on a curve equals the derivative of the curve at that point. Suppose the point has input $x_0$ and output $y_0=f(x_0)$, with $x_0$ in the same unit as $x$ and $y_0$ in the same unit as $y$. The gradient is

The tangent equation is

A reliable method is to find the point first. Then differentiate, evaluate the derivative at that point, and substitute into the line equation. The derivative expression by itself is not a tangent equation.

Normal equations

A normal is a line through a point on a curve that is perpendicular to the tangent there. If $m_t\ne 0$ on an equally scaled Cartesian plane, the normal gradient is

Its equation is

When the tangent is horizontal, the normal is the vertical line $x=x_0$; a finite normal gradient cannot be used. A vertical tangent also needs separate geometric interpretation.

Graphing technology can draw a tangent directly or estimate its gradient numerically. Check that the line goes through the stated point and has the expected orientation. Analytic differentiation still matters because it gives the exact gradient and makes any assumptions visible.

Tangents can model instantaneous velocity and price elasticity. Normals occur in ray optics and in directions perpendicular to equipotential curves. Technology has made both easier to generate and share, but an automatically drawn tangent isn’t self-justifying knowledge. Its mathematical meaning comes from the derivative and the assumptions behind the display.

Finding zero-gradient points

A stationary point is a point on a differentiable curve at which the first derivative is zero. Its input must satisfy

Find $f'(x)$ algebraically, or use a GDC or graphing package to generate it. Next, solve $f'(x)=0$. Substitute every solution into $f$ to get the full coordinates; giving only the input leaves out the point’s output.

A local maximum is a point whose function value is at least as large as nearby function values. By contrast, a local minimum is a point whose function value is at most as large as nearby function values. At a differentiable turning point, look at how the derivative changes sign:

• positive to negative gives a local maximum;
• negative to positive gives a local minimum;
• no sign change means the stationary point is not a local extremum.

Sign chart for local and global extrema.

Local is not global

A local extremum is compared only with nearby points. A global maximum is a point attaining the greatest function value over the entire stated domain, whereas a global minimum is a point attaining the least function value over that domain. An endpoint or a different part of the graph may beat a local turning point.

In an application, a stationary point might represent maximum profit, minimum cost, or an extreme area or volume. Zero-gradient and turning-point ideas also describe physical changes on displacement–time and velocity–time graphs, including oscillatory motion.

The graphing window can change what seems to be the “whole” behaviour of a model. A local maximum, for example, may appear global until the domain is widened. This raises a broader knowledge issue: mathematical descriptions don’t simply record the world. Our choices of variables, scales and domains shape the description we obtain.

Building an optimisation model

Optimisation is a modelling process that finds the greatest or least feasible value of a chosen quantity. That quantity is the objective, while the restrictions given by the context become the constraints.

A reliable method is:

1. Define the variables and give their units.
2. Write a formula for the quantity being maximised or minimised.
3. Apply the constraint to rewrite that quantity as a function of one variable.
4. Use the context to state the feasible domain.
5. Differentiate, then solve $f'(x)=0$ to find feasible stationary values.
6. Compare the relevant stationary values with the domain boundaries.
7. State the requested dimensions or quantities, including units and a conclusion that fits the context.

Context affects the mathematics directly. Lengths must be positive. Production may have to be an integer, and any stationary solution outside the feasible domain must be rejected.

Common applications include maximising profit or minimising cost. Optimisation can also maximise the capacity of packaging made from a fixed amount of material, or reduce material while preserving a required capacity. Kinematic optimisation provides a useful physical connection, but it is not assessed in standard-level examinations under this statement.

Interpreting the optimum

Solving $f'(x)=0$ gives candidates; it does not automatically give the answer. Check that the required maximum or minimum has been found by considering the derivative’s sign, the graph, or a comparison of feasible values. For a closed domain, check the endpoints too.

Optimisation can reveal value judgements. For example, a tax on disposable containers may change the cost function, making low-material or reusable designs economically preferable. Mathematics identifies the optimum under the assumptions given. It cannot decide whether the objective should be profit, resource use, access or environmental damage. Similar tensions appear in allocative efficiency in economics.

Standard derivatives

Trigonometric derivatives require angles to be measured in radians. The results you need are

where $\sec x=\frac{1}{\cos x}$. Both the inputs and outputs of these trigonometric functions are dimensionless.

For exponential and logarithmic functions, use

where $e$ is the dimensionless natural exponential base. For the real logarithmic formula, $x>0$ is required. Rational powers follow the rule

where $\mathbb{Q}$ is the set of rational numbers. Always respect the real domain of the original power function, particularly when roots or negative exponents are involved.

Chain rule

The chain rule is a differentiation rule that differentiates a composite function by multiplying the outer derivative by the inner derivative. Suppose $y=f(u)$ and $u=g(x)$, with $u$ as an intermediate quantity measured in the unit required by the composite model and $g$ as the inner function. Then

The equivalent function notation is

The factor $g'(x)$ is often missed. For example,

Product and quotient rules

The product rule is a differentiation rule that finds the derivative of two multiplied differentiable functions. Let $u=u(x)$ and $v=v(x)$, where $v$ is a second differentiable quantity with context-dependent SI units. The rule is

You need both terms. Simply differentiating each factor and multiplying the two answers is incorrect.

The quotient rule is a differentiation rule that finds the derivative of one differentiable function divided by another non-zero differentiable function:

Keep the numerator in the correct order: denominator times the derivative of the numerator, minus numerator times the derivative of the denominator.

You can combine these rules. First identify the outermost structure. A product, for instance, may include a composite factor; a quotient may involve trigonometric or exponential functions. A symbolic differentiator can check the final expression, but simplify with care because equivalent derivatives may look very different.

Related rates

A related-rates problem is a rate-of-change problem in which an equation links quantities that vary with the same independent variable. Usually, the independent variable is time. Begin with the equation linking the quantities. Differentiate the entire equation with respect to time before substituting the instantaneous measurements.

For a circular area,

where $A$ is area ($\text{m}^2$) and $\pi$ is the dimensionless circle constant. Differentiating with respect to time gives

Here $\frac{dA}{dt}$ is the area rate ($\text{m}^2\text{ s}^{-1}$), while $\frac{dr}{dt}$ is the radial rate ($\text{m s}^{-1}$). The signs matter: when the radius is shrinking, its radial rate is negative.

These techniques build on earlier work with extrema and optimisation. They also support models such as uniform circular motion and induced emf. Historically, powerful calculations sometimes came before a fully rigorous foundation. Euler achieved major results before the later formal work associated with Cauchy. Mathematical progress can move between productive intuition and symbolic technique, with justification following later, rather than developing in one fixed sequence.

The second derivative

A second derivative is the derivative of the first derivative and therefore measures how the first rate of change itself changes. It has two standard notations:

When $x$ and $y$ have units, the second derivative’s units are the output unit divided by the square of the input unit. Take a displacement model: its first derivative gives velocity, while its second derivative gives acceleration. This connects second derivatives to kinematics, simple harmonic motion and second-order differential equations, though those later topics aren’t needed here.

The second derivative test

At a stationary input where $f'(x)=0$:

• if $f''(x)>0$, the point is a local minimum;
• if $f''(x)<0$, the point is a local maximum;
• if $f''(x)=0$, the test is inconclusive.

The geometry helps explain these signs. A positive second derivative shows that the gradient is increasing, which produces a bowl-like shape. A negative second derivative shows that the gradient is decreasing, giving a cap-like shape.

Concavity and inflexion

A function is concave-up on an interval when its gradients increase throughout that interval, corresponding to $f''(x)>0$. It is concave-down on an interval when its gradients decrease throughout that interval, corresponding to $f''(x)<0$.

A point of inflexion is a point on a curve at which the concavity changes. Confirm this by checking whether $f''$ changes sign across the point. Solving $f''(x)=0$ only identifies candidates; it doesn’t prove there is an inflexion, since the second derivative may touch zero without changing sign.

In context, an inflexion shows a change in how a rate develops. Growth, for example, may continue but shift from accelerating to decelerating. The output doesn’t need to have a maximum or minimum at that point.

Mathematics can describe features of oscillations and even musical waveforms. Still, representing music mathematically doesn’t make the lived experience of music identical to an equation. A model records selected structures, such as frequency, amplitude or change. Whether this makes music “mathematical” depends on what we mean by expressing and explaining a phenomenon.