Calculus

The One Thing

Calculus uses limits to define derivatives, which measure instantaneous rate of change and give the slope of a curve. Differentiation then supports the analysis of motion, growth, increasing and decreasing behavior, stationary points, and optimization.

Definitions and Results

• Function: A rule that assigns exactly one output to each allowed input.
• Limit: The value that a function approaches when its input approaches a specified number. The notation is ${\lim }_{x\to a}f(x)$.
• Polynomial limit: For a polynomial function, ${\lim }_{x\to a}f(x)=f(a)$.
• Continuity: A function is continuous at $x=a$ when ${\lim }_{x\to a}f(x)=f(a)$.
• Derivative: The instantaneous rate of change of a function with respect to its variable.
• First Principle of Differentiation:

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

• Geometrical Meaning of Derivative: The derivative at a point gives the slope of the tangent to the graph at that point.
• Tangent: A line that touches a curve at a point and has the same slope as the curve there.
• Differentiability: A function is differentiable at a point if its derivative exists at that point.
• Differentiability and continuity: Differentiability at a point implies continuity at that point, but continuity does not always imply differentiability.
• Derivative of a constant:

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

• Power Rule:

$$\frac{d(x^n)}{dx}=n{x}^{n-1},$$ where $n$ is a suitable real number.

• Derivative of $x$:

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

• Constant Multiple Rule:

$$\frac{d}{dx}[kf(x)]=k{f}^{\prime }(x).$$

• Sum and Difference Rules:

$$\frac{d}{dx}[u\pm v]=\frac{du}{dx}\pm \frac{dv}{dx}.$$

• Product Rule: For $y=u\cdot v$,

$$\frac{dy}{dx}=u\frac{dv}{dx}+v\frac{du}{dx}.$$

• Quotient Rule: For $y=u/v$,

$$\frac{dy}{dx}=\frac{v(du/dx)-u(dv/dx)}{v^2},$$ provided $v\ne 0$.

• Chain Rule: For $y=f(g(x))$,

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

• Exponential derivatives:

$$\frac{d(e^x)}{dx}=e^x,\frac{d(a^x)}{dx}=a^x\ln (a),$$ for $a>0$ and $a\ne 1$.

• Logarithmic derivative:

$$\frac{d(\ln x)}{dx}=\frac{1}{x},$$ for $x>0$.

• Basic trigonometric derivatives:

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

• Derivative of tangent:

$$\frac{d(\tan x)}{dx}={\sec }^{2}x,$$ wherever $\tan x$ is defined.

• Radians: Angles in trigonometric differentiation are normally measured in radians.
• Tangent slope: The slope of the tangent to $y=f(x)$ at $x=a$ is $f^{\prime }(a)$.
• Equation of the tangent: At $(a,f(a))$,

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

• Equation of the normal: At $(a,f(a))$,

$$y-f(a)=-\left[\frac{1}{f^{\prime }(a)}\right](x-a),$$ when $f^{\prime }(a)\ne 0$.

• Rate of Change: The change in one quantity compared with the change in another; a derivative gives the instantaneous rate of change.
• Motion: If $s(t)$ represents displacement, then

$$v(t)=\frac{ds}{dt},a(t)=\frac{dv}{dt}=\frac{d^{2}s}{d{t}^2}.$$

• Increasing Function: A function generally increases on an interval where its derivative is positive.
• Decreasing Function: A function generally decreases on an interval where its derivative is negative.
• Critical Point: A point where the derivative is zero or does not exist, often considered when locating maximum or minimum values.
• Stationary point: A point where the derivative is zero. Further checking is needed to determine whether it is a maximum, minimum, or neither.
• Local Maximum: A function value that is greater than nearby function values.
• Local Minimum: A function value that is smaller than nearby function values.
• Optimization: The process of finding the greatest or least possible value of a quantity under given conditions.

Worked Methods

1. Evaluating a limit

• Identify the limiting form ${\lim }_{x\to a}f(x)$.
• For a polynomial function, use direct substitution:

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

• Interpret the result as the value approached by $f(x)$ as $x$ approaches $a$.

2. Testing continuity at $x=a$

• Find ${\lim }_{x\to a}f(x)$.
• Find $f(a)$.
• Compare the two values.
• Conclude that the function is continuous at $x=a$ precisely when

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

3. Differentiating from first principles

• Start with

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

• Substitute the expression for $f(x+h)$.
• Subtract $f(x)$.
• Simplify and cancel any factor of $h$, where valid.
• Take the limit as $h\to 0$.

This method defines the derivative and establishes the connection between limits and instantaneous rate of change.

4. Applying basic differentiation rules

• Rewrite the expression into identifiable terms where necessary.
• Apply the derivative of a constant, the derivative of $x$, and the power rule.
• Apply the constant multiple rule term by term.
• Apply the sum and difference rules.
• Simplify the resulting derivative.

The principal rules are $$\frac{d(c)}{dx}=0,\frac{d(x)}{dx}=1,\frac{d(x^n)}{dx}=n{x}^{n-1},$$ $$\frac{d}{dx}[kf(x)]=k{f}^{\prime }(x),\frac{d}{dx}[u\pm v]=\frac{du}{dx}\pm \frac{dv}{dx}.$$

5. Differentiating a product

For $y=u\cdot v$:

• Identify $u$ and $v$.
• Find $du/dx$ and $dv/dx$.
• Substitute into

$$\frac{dy}{dx}=u\frac{dv}{dx}+v\frac{du}{dx}.$$

• Simplify.

6. Differentiating a quotient

For $y=u/v$:

• Identify the numerator $u$ and denominator $v$.
• Find $du/dx$ and $dv/dx$.
• Substitute into

$$\frac{dy}{dx}=\frac{v(du/dx)-u(dv/dx)}{v^2}.$$

• Confirm that $v\ne 0$.
• Simplify.

7. Differentiating a composite function

For $y=f(g(x))$:

• Identify the outer function $f$ and inner function $g(x)$.
• Differentiate the outer function with respect to its argument.
• Multiply by the derivative of the inner function:

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

• Simplify.

8. Finding the tangent and normal

At $x=a$:

• Calculate $f(a)$, giving the point $(a,f(a))$.
• Calculate $f^{\prime }(x)$.
• Evaluate $f^{\prime }(a)$, the tangent slope.
• Use the tangent equation

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

• If $f^{\prime }(a)\ne 0$, use the normal equation

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

9. Analyzing motion

Given displacement $s(t)$:

• Differentiate $s(t)$ with respect to $t$ to obtain velocity:

$$v(t)=\frac{ds}{dt}.$$

• Differentiate velocity to obtain acceleration:

$$a(t)=\frac{dv}{dt}=\frac{d^{2}s}{d{t}^2}.$$

• Interpret the results as instantaneous rates of change.

10. Determining increasing and decreasing intervals

• Find $f^{\prime }(x)$.
• Locate points where $f^{\prime }(x)=0$ or where $f^{\prime }(x)$ does not exist.
• Use these points to divide the domain into intervals.
• Determine the sign of $f^{\prime }(x)$ on each interval.
• Conclude that the function generally increases where $f^{\prime }(x)>0$ and decreases where $f^{\prime }(x)<0$.

11. Classifying stationary points

• Find $f^{\prime }(x)$.
• Solve $f^{\prime }(x)=0$.
• Check the behavior of $f^{\prime }(x)$ on either side of each stationary point.
• Use the change in sign, together with relevant function values, to determine whether the point is a local maximum, local minimum, or neither.

A derivative equal to zero identifies a stationary point but does not by itself determine its classification.

12. Solving an optimization problem

• Identify the variable.
• Form the required function subject to the given conditions.
• Differentiate the function.
• Find critical points where the derivative is zero or does not exist.
• Compare relevant function values.
• Select the greatest or least value required by the problem.

Applications include maximizing profit, minimizing cost, or finding the greatest area under constraints.

Where It Goes Wrong

• Treating a limit as automatically equal to the function value without checking the required continuity condition ${\lim }_{x\to a}f(x)=f(a)$.
• Omitting the limit $h\to 0$ or simplifying incorrectly when using

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

• Forgetting that the denominator in the Quotient Rule must be nonzero.
• Applying a product, quotient, or Chain Rule expression as though it were a sum of separate terms.
• Assuming that $f^{\prime }(x)=0$ automatically gives a maximum or minimum; further checking is required.
• Using trigonometric differentiation without observing that angles are normally measured in radians, or using the normal equation when $f^{\prime }(a)=0$.

What Gets Asked

This material supports questions requiring students to:

• Evaluate limits, including polynomial limits, and test continuity at $x=a$.
• Derive or apply the First Principle of Differentiation.
• Differentiate constants, powers, exponential functions, logarithms, and trigonometric functions.
• Apply the constant multiple, sum and difference, Product, Quotient, and Chain Rules.
• Find the slope and equation of a tangent, and the equation of a normal.
• Interpret derivatives as rates of change.
• Determine velocity and acceleration from displacement $s(t)$.
• Identify intervals on which a function is increasing or decreasing.
• Locate critical and stationary points and classify local maxima and local minima.
• Form and solve optimization problems by comparing relevant values.