Calculus

The One Thing

Limits describe the value a function approaches, while derivatives measure instantaneous rate of change. Derivatives therefore connect algebraic formulas with tangent slopes, increasing and decreasing behaviour, stationary points, rates such as velocity, and basic optimization.

Definitions and Results

• Limit: The number that $f(x)$ approaches as $x$ approaches a specified value, written as ${\lim }_{x\to a}f(x)$.
• Left-hand limit: The value approached by $f(x)$ as $x$ approaches $a$ from values less than $a$, written as ${\lim }_{x\to a^{-}}f(x)$.
• Right-hand limit: The value approached by $f(x)$ as $x$ approaches $a$ from values greater than $a$, written as ${\lim }_{x\to a^{+}}f(x)$.
• Existence of a limit: A limit exists at $x=a$ only when

$$\underset{x\to a^{-}}{\lim }f(x)=\underset{x\to a^{+}}{\lim }f(x).$$

• Continuity: A function is continuous at $x=a$ if $f(a)$ exists, ${\lim }_{x\to a}f(x)$ exists, and

$$\underset{x\to a}{\lim }f(x)=f(a).$$ For a continuous function, direct substitution gives ${\lim }_{x\to a}f(x)=f(a)$, provided the substituted value is defined.

• Derivative: The derivative gives the instantaneous rate of change of a function and is written as $f^{\prime }(x)$ or $\frac{dy}{dx}$.
• First principle of differentiation: When the limit exists,

$$f^{\prime }(x)=\underset{h\to 0}{\lim }\frac{f(x+h)-f(x)}{h}.$$ At $x=a$, $$f^{\prime }(a)=\underset{h\to 0}{\lim }\frac{f(a+h)-f(a)}{h}.$$

• Geometrical meaning of derivative: At a point on a curve, the derivative is the slope of the tangent to the curve.
• Tangent: A straight line that touches a curve at a point and has slope equal to the derivative at that point.
• Normal: A line perpendicular to the tangent at a point on a curve.
• Differentiability: A function is differentiable at a point if its derivative exists there. Differentiability implies continuity, although continuity does not necessarily imply differentiability.
• Derivative of a constant:

$$\frac{d(c)}{dx}=0.$$

• Power rule: For a constant $n$,

$$\frac{d(x^n)}{dx}=n{x}^{n-1}.$$

• Product rule: For functions $u$ and $v$,

$$\frac{d(uv)}{dx}=u\frac{dv}{dx}+v\frac{du}{dx}.$$

• Quotient rule: For functions $u$ and $v$, with $v\ne 0$,

$$\frac{d}{dx}\left(\frac{u}{v}\right)=\frac{v\frac{du}{dx}-u\frac{dv}{dx}}{v^2}.$$

• Chain rule: For a composite function $y=f(g(x))$,

$$\frac{dy}{dx}=f^{\prime }(g(x))g^{\prime }(x).$$

• Standard trigonometric derivatives:

$$\frac{d}{dx}(\sin x)=\cos x,\frac{d}{dx}(\cos x)=-\sin x,\frac{d}{dx}(\tan x)={\sec }^{2}x.$$

• Standard exponential and logarithmic derivatives:

$$\frac{d}{dx}(e^x)=e^x,\frac{d}{dx}(\log x)=\frac{1}{x},$$ where $\log x$ denotes the natural logarithm.

• Standard limits: With angles measured in radians,

$$\underset{x\to 0}{\lim }\frac{\sin x}{x}=1,\underset{x\to 0}{\lim }\frac{\tan x}{x}=1,\underset{x\to 0}{\lim }\frac{1-\cos x}{x}=0.$$

• Equation of the tangent: At $(x_1,y_1)$,

$$y-y_1=f^{\prime }(x_1)(x-x_1).$$

• Slope and equation of the normal: If $f^{\prime }(x_1)\ne 0$, the normal has slope

$$-\frac{1}{f^{\prime }(x_1)}$$ and equation $$y-y_1=-\left[\frac{1}{f^{\prime }(x_1)}\right](x-x_1).$$

• Stationary point: A point where the first derivative is zero, so the tangent is horizontal.
• Increasing function: A function is increasing on an interval when its derivative is positive throughout that interval.
• Decreasing function: A function is decreasing on an interval when its derivative is negative throughout that interval.
• Maximum and minimum: A maximum is a highest nearby value and a minimum is a lowest nearby value. They may occur at stationary points or endpoints.
• Stationary-point classification: If $f^{\prime }(x)$ changes from positive to negative, the stationary point is a local maximum. If $f^{\prime }(x)$ changes from negative to positive, it is a local minimum. The second derivative may also be used where appropriate.
• Rate of change: A derivative can represent quantities such as velocity, growth rate, or the slope of a graph. Its units are output units per input unit.

Worked Methods

Evaluating limits

• Identify the limiting value $a$ and the expression ${\lim }_{x\to a}f(x)$.
• Check the left-hand and right-hand behaviour. The limit exists only if

$$\underset{x\to a^{-}}{\lim }f(x)=\underset{x\to a^{+}}{\lim }f(x).$$

• If $f$ is continuous and direct substitution gives a defined value, substitute $x=a$:

$$\underset{x\to a}{\lim }f(x)=f(a).$$

• If direct substitution produces an indeterminate form such as $0/0$, factorise the rational expression and cancel the common factor where permitted.
• For trigonometric limits near zero, use the standard results

$$\underset{x\to 0}{\lim }\frac{\sin x}{x}=1,\underset{x\to 0}{\lim }\frac{\tan x}{x}=1,\underset{x\to 0}{\lim }\frac{1-\cos x}{x}=0,$$ with angles in radians.

Finding a derivative from first principles

• Start with

$$f^{\prime }(x)=\underset{h\to 0}{\lim }\frac{f(x+h)-f(x)}{h}.$$

• Substitute the specified function for $f(x+h)$ and $f(x)$.
• Simplify the numerator and cancel any factor of $h$, if possible.
• Evaluate the remaining limit as $h\to 0$.
• At a particular point $x=a$, use

$$f^{\prime }(a)=\underset{h\to 0}{\lim }\frac{f(a+h)-f(a)}{h}.$$

• Interpret the result as the instantaneous rate of change or the slope of the tangent.

Differentiating algebraic and composite functions

• Identify the structure of the function.
• Apply the derivative of a constant, the power rule, the product rule, the quotient rule, or the chain rule as appropriate.
• For $y=x^n$, use

$$\frac{dy}{dx}=n{x}^{n-1}.$$

• For a product $uv$, use

$$\frac{d(uv)}{dx}=u\frac{dv}{dx}+v\frac{du}{dx}.$$

• For a quotient $u/v$, use

$$\frac{d}{dx}\left(\frac{u}{v}\right)=\frac{v\frac{du}{dx}-u\frac{dv}{dx}}{v^2},$$ ensuring that $v\ne 0$.

• For a composite function $y=f(g(x))$, differentiate the outer function and multiply by the derivative of the inner function:

$$\frac{dy}{dx}=f^{\prime }(g(x))g^{\prime }(x).$$

• Use the standard derivatives for $\sin x$, $\cos x$, $\tan x$, $e^x$, and $\log x$ where required.

Finding the tangent and normal

• Identify the point $(x_1,y_1)$ on the curve.
• Differentiate the curve to obtain $f^{\prime }(x)$.
• Evaluate $f^{\prime }(x_1)$ to obtain the tangent slope.
• Substitute into

$$y-y_1=f^{\prime }(x_1)(x-x_1)$$ to obtain the tangent equation.

• If $f^{\prime }(x_1)\ne 0$, calculate the normal slope:

$$-\frac{1}{f^{\prime }(x_1)}.$$

• Use

$$y-y_1=-\left[\frac{1}{f^{\prime }(x_1)}\right](x-x_1)$$ to obtain the normal equation.

Finding stationary points and determining their nature

• Differentiate the function to find $f^{\prime }(x)$.
• Solve

$$f^{\prime }(x)=0.$$

• Substitute the resulting $x$-values into the original function to obtain the stationary points.
• Determine the nature of each point using a sign test or the second derivative where appropriate.
• A change in $f^{\prime }(x)$ from positive to negative indicates a local maximum.
• A change in $f^{\prime }(x)$ from negative to positive indicates a local minimum.

Determining increasing and decreasing intervals

• Find $f^{\prime }(x)$.
• Identify critical values, including solutions of $f^{\prime }(x)=0$.
• Test the sign of $f^{\prime }(x)$ on each relevant interval.
• Where $f^{\prime }(x)>0$, the function is increasing.
• Where $f^{\prime }(x)<0$, the function is decreasing.

Solving an optimization problem

• Define the quantity to be optimized.
• Express that quantity in one variable.
• Differentiate the resulting function.
• Find critical points by solving $f^{\prime }(x)=0$.
• Determine whether the critical points give maxima or minima.
• Compare relevant values, including endpoints.
• Interpret the result in the context of the problem, including the units of the derivative or optimized quantity where relevant.

Where It Goes Wrong

• Treating a limit as existing without checking that the left-hand and right-hand limits are equal.
• Substituting directly into a rational expression that produces $0/0$, instead of factorising and cancelling where appropriate.
• Applying the standard trigonometric limits without remembering that the angles must be measured in radians.
• Confusing continuity with differentiability: a function may be continuous without being differentiable, whereas differentiability implies continuity.
• Using the quotient rule incorrectly or forgetting the condition $v\ne 0$.
• Finding $f^{\prime }(x)=0$ but failing to classify the stationary point, compare endpoints in optimization, or account for the normal formula requiring $f^{\prime }(x_1)\ne 0$.

What Gets Asked

• Evaluate a limit using direct substitution, one-sided limits, factorisation and cancellation, or the standard trigonometric limits.
• Determine whether a limit exists by comparing left-hand and right-hand limits.
• Test whether a function is continuous at $x=a$.
• Use the first principle of differentiation to derive $f^{\prime }(x)$ or calculate $f^{\prime }(a)$.
• Differentiate functions using the constant, power, product, quotient, and chain rules.
• Differentiate $\sin x$, $\cos x$, $\tan x$, $e^x$, and $\log x$.
• Find the equation of a tangent or normal at a specified point.
• Find stationary points and classify them using a sign test or the second derivative.
• Determine intervals on which a function is increasing or decreasing.
• Interpret a derivative as a rate of change, such as velocity, growth rate, or graph slope, including its units.
• Solve introductory maximum-minimum and optimization problems by comparing critical points and endpoints.