Limits: the idea, not the full machinery

A limit is the value a function’s output approaches when its input moves towards a chosen value. At this stage, you’ll estimate limits from tables of values or graphs. Formal analytic methods for calculating limits are not required here.

For a function $f(x)$, where $f$ is a real-valued function (output units depend on the context) and $x$ is the independent variable (unitless unless the context gives units), we write

The wording matters: $x$ approaches $a$, but it doesn’t have to equal $a$.

When possible, a table should approach the target from both sides. On a graph, follow the curve as $x$ moves towards that value. Technology works well for this kind of exploration. Spreadsheets, graphing software and a GDC allow you to zoom or tabulate, then test a guess before accepting it.

Values of $f(x)=x+1$ near $a=1$ approach $L=2$ from both sides.

Average rate, instantaneous rate, and the derivative

An average rate of change is a quotient comparing the total change in one variable with the total change in another over an interval. For a function between $x=a$ and $x=b$, it is

This quotient gives the gradient of the secant line through the two points.

A derivative is a function or value giving the instantaneous rate of change of one variable with respect to another. Geometrically, it gives the gradient of the tangent to the curve. Depending on the variables used in the question, the first derivative may be written as $\frac{dy}{dx}$, $f'(x)$, $\frac{dV}{dr}$ or $\frac{ds}{dt}$. Here $y$ is the dependent variable (same unit as the function output), $V$ is volume (for example m$^3$), $r$ is radius (m), $s$ is displacement (m), and $t$ is time (s).

The basic picture is simple: as two points move closer and closer together, a secant gradient becomes a tangent gradient. Calculus is therefore a limiting process. That makes it useful in physics, economics and chemistry, where velocity, marginal cost, marginal revenue, induced emf and gradients of experimental curves are all rates of change in disguise.

Historically, the language of limits is closely linked to difficult questions about zero and infinitesimally small quantities. This isn’t just trivia. It shows that mathematical intuition can be powerful, but careful definitions are still needed when that intuition is pushed to the edge.

Reading the sign of the derivative

An increasing function has output values that rise as the input values move from left to right over an interval. For a decreasing function, the output values fall as the input values move from left to right over an interval.

The derivative shows the local direction of travel:

• if $f'(x)>0$ on an interval, the function is increasing there;
• if $f'(x)<0$ on an interval, the function is decreasing there;
• if $f'(x)=0$ at a point, the tangent is horizontal there.

Be careful with the last line. A horizontal tangent doesn't automatically give a maximum or a minimum. The point could be a turning point or a stationary point of inflexion.

Sign tables

First solve $f'(x)=0$. Then place the critical $x$-values in increasing order and test the sign of $f'(x)$ in each interval. If the original function is undefined at any $x$-value, use that value as a divider as well; the sign can change across a break in the domain.

For example, the sign pattern $+$, $0$, $-$ shows that the graph rises and then falls, giving a local maximum. With $-$, $0$, $+$, the graph falls and then rises, so there is a local minimum. The pattern $+$, $0$, $+$ shows that the function is increasing on both sides, despite having a horizontal tangent at the middle point.

The power rule for integer powers

For integer powers, use the rule

Bring the power down as a multiplier, then reduce the power by one.

Apply the rule to each term in polynomial-style expressions such as

Constants, sums and negative powers

A constant multiple remains in front. For a sum, differentiate each term separately:

When possible, rewrite roots or fractions as powers before differentiating. For example, $\sqrt{x}=x^{\frac12}$ is not covered by the integer version in this statement. Still, rewriting in power form becomes essential in the next differentiation statement.

Tangent gradients

A tangent is a straight line with the same gradient as a curve at the point where the line touches it.

The straight-line form you’ll usually need is

For an analytic approach, differentiate first. Then substitute the $x$-value, find the point and write the equation of the line. On technology papers, a GDC or graphing package can also draw the tangent and provide its equation, but you still need to understand what the calculator is reporting.

Normal gradients

A normal is a straight line perpendicular to the tangent at the point of contact.

provided $m\ne0$. When the tangent is horizontal, the normal is vertical, so its equation is $x=p$.

A final answer for a tangent or normal must be an equation of a straight line, not just a gradient. The same idea appears in applications such as instantaneous velocity or price elasticity: the derivative gives a local linear model of the curve.

Standard derivatives

The standard derivatives needed here are

You’ll also need

For the trigonometric derivatives, $x$ must be measured in radians. This convention isn’t just cosmetic: the formulae above depend on radian measure.

Sums and constant multiples work in the usual way:

The chain rule

A composite function is formed when one function is applied to the output of another.

Then

In classroom language: differentiate the outside, leave the inside alone, and multiply by the derivative of the inside. For example, $\sin(3x-1)$ differentiates to $3\cos(3x-1)$, not just $\cos(3x-1)$.

Nested functions and chain rule factors

Product and quotient rules

The product rule gives the derivative of two functions multiplied together:

Don’t multiply the two derivatives. That’s the classic wrong turn.

The quotient rule gives the derivative of one function divided by another:

The connection with composite functions matters because many expressions require more than one rule. A quotient might need the chain rule in its denominator, for instance, while a product could have $e^{x^2}$ as one factor.

What the second derivative measures

The second derivative comes from differentiating the first derivative. We write it as

It shows how the gradient changes. When $f'(x)$ is increasing, the curve bends upwards. When $f'(x)$ is decreasing, it bends downwards.

A graph is concave-up on an interval if $f''(x)>0$ throughout that interval. It is concave-down on an interval if $f''(x)<0$ throughout that interval.

Linking the graphs of $f$, $f'$ and $f''$

Think of the graph of $f'$ as the gradient graph of $f$. At a horizontal tangent on $f$, $f'(x)=0$. If $f$ is increasing, $f'$ lies above the $x$-axis; if $f$ is decreasing, $f'$ lies below the $x$-axis.

In the same way, $f''$ is the gradient graph of $f'$. A turning point on $f'$ gives $f''(x)=0$. A positive $f''$ shows that $f'$ is increasing, while a negative $f''$ shows that $f'$ is decreasing.

Technology helps here. Plot $f$, $f'$ and $f''$ on the same axes, then check the vertical alignments. Turning points of $f$ match zeros of $f'$. Changes in the concavity of $f$ match zeros and sign changes of $f''$.

Local maximum and minimum points

A local maximum point occurs where the output is greater than the nearby output values. At a local minimum point, the output is less than the nearby output values.

A stationary point lies on a differentiable curve and satisfies $f'(x)=0$. A turning point is a stationary point where the function switches from increasing to decreasing, or from decreasing to increasing.

There are two standard tests:

• First derivative sign test: a change in $f'$ from $+$ to $-$ gives a local maximum; a change from $-$ to $+$ gives a local minimum.
• Second derivative test:

, there is a local minimum at $x=c$

If $f'(c)=0$ and $f''(c)<0$, there is a local maximum.

When $f'(c)=0$ and $f''(c)=0$, the second derivative test has failed. It hasn’t proved anything, so go back to a sign test.

Optimisation

Optimisation means finding the input value that makes a quantity as large or as small as possible under given conditions. Typical syllabus examples include profit, area and volume. The same mathematics can model cost, efficiency, travel time and many physical quantities.

For a clean optimisation solution, first define the quantity to optimise. Use the constraints to write it in one variable, then differentiate and solve for the candidates. Test those candidates and interpret the answer in context. If the result gives a box a negative length or says that volume is maximised at zero size, re-check the modelling.

Points of inflexion

A point of inflexion occurs where a curve changes concavity: from concave-up to concave-down, or from concave-down to concave-up. At such a point, $f''(x)=0$ and $f''$ changes sign.

The word and matters here. The condition $f''(x)=0$ on its own is not enough. For example, $y=x^4$ has $f''(0)=0$, but the curve is concave-up on both sides of $(0,0)$. There is no point of inflexion there.

A point of inflexion can have either zero gradient or non-zero gradient. If $f'(x)=0$ as well, it has a horizontal tangent. If $f'(x)\ne0$, the tangent is sloping. Both types are in the syllabus.

Continuity and differentiability

A continuous function at a point has a graph with no break, jump or hole at that input value. Informally, you could draw the graph near the point without lifting your pen.

A differentiable function at a point has a well-defined tangent gradient at that input value. Differentiability is a stronger condition than continuity: differentiable implies continuous, but continuous does not imply differentiable. For example, a graph with a sharp corner may be continuous there but not differentiable.

In examinations, you won't be asked to carry out formal tests for continuity or differentiability. Instead, you need enough understanding of both ideas to interpret graphs and limits.

Four common point behaviors: smooth, corner, jump, and hole.

Limits, convergence and divergence

A convergent limit occurs when the function values approach a finite value. With a divergent limit, the function values don't settle to a finite value; they may grow without bound or oscillate.

The same language applies naturally to infinite geometric sequences. A sequence may approach a finite value, or it may fail to settle. Limits of functions follow the same broad idea.

First principles

The derivative from first principles is

The fraction gives the gradient of a secant line. Taking the limit turns this into the gradient of the tangent line.

For this syllabus point, use the definition for polynomials only. A safe algebraic routine is to substitute $x+h$ and expand carefully. Then subtract $f(x)$, factor and cancel $h$, and only then let $h\to0$. Don't set $h=0$ at the start, as that would involve dividing by zero.

Higher derivatives

A higher derivative comes from differentiating repeatedly. The notation

For example, $f^{(3)}(x)$ is the third derivative. Patterns involving repeated differentiation also provide a natural setting for proof by mathematical induction.

Indeterminate quotients

An indeterminate form is an expression whose form alone does not determine a limit. The forms used here are

You will encounter limits such as

where $g(x)$ is a real-valued denominator function, with the same output unit as needed for the quotient. When direct substitution produces $0/0$ or $\infty/\infty$, l'Hopital's rule may be used:

provided its conditions are satisfied and the new limit exists.

Differentiate the numerator and denominator separately—this is not the quotient rule. If the result is still indeterminate, apply l'Hopital's rule again.

Limits at infinity and horizontal asymptotes

A limit such as $\lim_{x\to\infty}\frac{f(x)}{g(x)}$ often describes the graph's end behaviour. When the quotient approaches a finite number, the graph may have a horizontal asymptote. This links limits at infinity to rational functions and asymptotes.

Maclaurin series as an alternative

Near $0$, a Maclaurin expansion can expose the first non-zero term in both the numerator and denominator. Consider the well-known result

This follows from the expansion $\sin\theta=\theta-\frac{\theta^3}{3!}+\cdots$, though other limiting arguments can also be used.

Keep enough terms in the series to identify the first one that doesn't cancel. If the numerator and denominator both begin with the same power of $x$, the coefficients of those leading terms usually determine the limit.

Implicit differentiation

An implicit equation relates variables without necessarily making one of them the subject. For example, the equation of a curve might contain $x$ and $y$ mixed together rather than expressing one directly in terms of the other.

With implicit differentiation, differentiate both sides with respect to $x$, treating $y$ as a function of $x$. Whenever you differentiate a term involving $y$, the chain rule introduces a factor of $\frac{dy}{dx}$.

For example, if a term contains $y^2$, then

Implicit differentiation can also be used to find derivatives of inverse functions, since an inverse relation can often be rewritten in a more convenient implicit form.

Related rates

A related rates problem involves two or more changing quantities connected by an equation. Differentiation then links their rates of change.

When both variables depend on time, differentiate with respect to $t$. For the area of a circle, for example,

Differentiate the equation before substituting values for the particular instant, unless a variable really is constant.

Optimisation, including endpoints

The optimisation methods from earlier still apply, though the model may now involve implicit differentiation or the chain rule. An optimum doesn’t always occur at a stationary point. On a closed interval, the candidates are the stationary points and the endpoints.

If a variable is restricted by

test $x=\alpha$, $x=\beta$, and every valid stationary point inside the interval. A stationary point may be a maximum even when the problem asks for a minimum, so the context doesn’t remove the need to check.

Further standard derivatives

Learn the extra derivative formulae as patterns you can quickly recognise:

The syllabus may use cosec instead of $\csc$. As a memory hook, the co-functions $\csc x$ and $\cot x$ both give expressions with a negative sign when differentiated.

For exponentials and logarithms with base $A$

For inverse trigonometric functions,

Use the chain rule when these formulae have linear inside functions. If the inside is $px+q$, where $p$ is the linear coefficient (inverse unit of $x$) and $q$ is a constant shift (unitless for trig and log inputs), multiply by $p$ during differentiation.

Indefinite integrals as families of curves

An indefinite integral is a family of antiderivative functions that differ by a constant. Here, this includes integrating the derivatives of the functions listed above, as well as composites with a linear function.

For example,

The factor $\frac12$ compensates for the inside derivative of $2$.

Similarly,

The phrase family of curves matters. Changing $C$ shifts the graph vertically but leaves its derivative unchanged.

Rearranging integrands using partial fractions

A partial fraction decomposition rewrites a rational expression as a sum of simpler rational expressions. This helps because the simpler fractions often integrate to logarithms or inverse trigonometric functions.

For example, once a quadratic denominator has been factorised, an expression of the form

can be rewritten as

The integral can then be found using logarithms:

Completing the square may instead reveal an arctangent form, such as an integrand based on $\frac{1}{(x+1)^2+4}$. Don’t force one method. Rearrange the expression until it matches a derivative you recognise.