Year11-MATH-3-1-2 Introduction to Differential Calculus

In Chapter 1, we looked at static relationships. In Chapter 2, we move into Calculus, the mathematics of change. Instead of asking “Where is the object?”, we begin to ask “How fast is the object moving at this exact moment?”

2.1 The Concept of a Gradient

In linear functions ($y=mx+c$), the gradient is constant. No matter where you are on the line, the “steepness” is the same. However, for curves like $f(x)=x^2$, the steepness changes constantly.

Secants vs. Tangents

• Average Rate of Change (Secant): The gradient of a line connecting two distinct points on a curve.
• Instantaneous Rate of Change (Tangent): The gradient of a line that just touches the curve at a single point. This is the core of calculus.

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2.2 The Derivative Function

The “Derivative” is simply a formula that tells us the gradient of the tangent at any value of x. We use two main types of notation:

1. Leibniz’s Notation: $\frac{dy}{dx}$​ (read as “the derivative of y with respect to x”).
2. Lagrange’s Notation: $f′(x)$ (read as “f prime of x”).

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2.3 The Power Rule

The Power Rule is the most vital shortcut in Mathematical Methods. It allows us to find the derivative of any polynomial function without using complex “first principles” limits.

The Power Rule

If $f(x)=x^n$, then:

$f′(x)=nx^{n−1}$

Steps to differentiate:

1. Multiply the coefficient by the current power ($n$).
2. Subtract 1 from the power.

A power Rule is a relationship in which one quantity is proportional to the power of another (expressed as ). It is a law widely observed in nature and social phenomena.

It is characterized by an asymmetric distribution in which a small number of factors have extremely large values ​​(such as influence or frequency) while many factors remain small. This is observed in the number of social media followers, the population of a city, the magnitude of earthquakes (the Gutenberg-Richter law), and income distribution (the Pareto principle).

Key Characteristics of the Power Rule

• Mathematical Expression: (where is a constant, and is a power exponent).

• Distribution Skew: There are a few “big” events and many “small” events.

• Linearity on a Log-Log Graph: When a graph is plotted on a log-log (log-log) scale, it appears as a linear relationship.

• Universality: Similar patterns can be observed in phenomena from different fields (such as physics, economics, and biology).

Familiar examples:

• Number of social media followers: Some celebrities have huge followings, while most people have small ones.

• Urban population: The population is concentrated in a few large cities, while many regional cities have small populations.

• Earthquake size: Small earthquakes are frequent, but large earthquakes are very rare.

• Word frequency: High-frequency words are rare, while low-frequency words are very common (known as Zipf’s law).

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In Australian Year 11 Mathematical Methods (and Specialists), the Power Rule is the foundational, most commonly used shortcut for finding the derivative of power functions ($x^n$). It allows students to calculate the gradient function ( $\frac{dy}{dx}$or $f^`(x)$) without using the laborious “first principles” limit definition every time. 

>> The Power Rule Formula 

If $f(x)=x^{n}$, then the derivative is : $\frac{d}{dx}(x^{n})=nx^{n-1}$

Where $n$ is any real number.

If the term has a constant coefficient $a$, such as $f(x)=ax^n$, the rule is: $\frac{d}{dx}(ax^{n})=anx^{n-1}$

Multiply the exponent by the coefficient, then subtract 1 from the exponent. 

>> Step-by-Step Application 

1. Identify $n$: Look at the power of $x$.
2. Bring it down: Multiply the entire term by this power ($n$).
3. Reduce it: Subtract 1 from the original power ($n – 1$). 

Example 1: Simple Power

$y = x^5$

$\frac{dy}{dx}$ = $5x^{5 – 1}$ = $5x^4$

Example 2: Coefficient and Power

$y = 4x^3$

$\frac{dy}{dx}$ = $(3×4) x^{3-1}$ = $12x^2$

>> Key Applications in Year 11

Polynomials: The power rule is applied to each term individually (Sum Rule).

• Example: $\frac{d}{dx}(3x^{2}+5x+2)=6x+5$.
• Negative Exponents (Reciprocals): Used for terms in the denominator.
• Example: $\frac{d}{dx}(\frac{1}{x^{2}})=\frac{d}{dx}(x^{-2})=-2x^{-3}=-\frac{2}{x^{3}}$.
• Fractional Exponents (Roots): Used for square roots, etc..
• Example: $\frac{d}{dx}(\sqrt{x})=\frac{d}{dx}(x^{1/2})=\frac{1}{2}x^{-1/2}=\frac{1}{2\sqrt{x}}$.
• Constant Rule: The derivative of a constant is 0.
• Example: $\frac{d}{dx}(10)=0$.
• Linear Terms: The derivative of is .
• Example: $\frac{d}{dx}(7x)=7$.

>> Key Requirements & Common Pitfalls

• Preparation: You must rewrite expressions with $x$ in the denominator or under a root sign using exponent rules ( $1/x^{n}=x^{-n}$ and $\sqrt[m]{x^{n}}=x^{n/m}$) before applying the power rule.
• Negative Numbers: Remember that $n – 1$ for a negative exponent makes it more negative (e.g., derivative of $x^{-2}$ is $-2x^{-3}$).
• Simplification: Always simplify the numerical coefficient after multiplying. 

Note: In the NSW HSC and other Australian jurisdictions, the power rule is typically introduced in Year 11 as the first step into formal differentiation, following the study of limits and first principles. 

Constants and Linear Terms

• Constants: The derivative of a constant (e.g., $f(x)=5$) is 0. (A flat line has no steepness).
• Linear Terms: The derivative of $f(x)=ax$ is simply $a$.

Worked Example 1: Differentiating a Polynomial Differentiate $f(x)=4x^3−2x^2+5x−7$.

1. Term 1 ($4x^3$): 3×4=12, then 3−1=2. Result: $12x^2$.
2. Term 2 (−$2x^2$): 2×−2=−4, then 2−1=1. Result: $−4x$.
3. Term 3 ($5x$): Gradient of $5x$ is simply 5.
4. Term 4 (−7): Derivative is 0.

Final Answer: $f′(x)=12x^2−4x+5$

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2.4 Finding the Gradient at a Point

Once you have the derivative function ($f′(x$)), you can find the exact steepness of the curve at any $x$-coordinate by substituting the value into the derivative.

Worked Example 2: Finding a Specific Gradient Find the gradient of the curve $y=x^2+3x$ at the point where $x$=2.

1. Find the derivative: $\frac{dy}{dx}$​ = $2x+3$.
2. Substitute x=2: 2(2)+3=7.
3. Interpretation: At the point (2,10), the curve is rising at a rate of 7 units up for every 1 unit across.

2.5 Practice Problems

Part A: Basic Differentiation

Differentiate the following with respect to $x$:

1. $f(x)=x^5$
2. $y=10x^2−4x$
3. $g(x)=\frac{1}{3}​x^3+5$
4. $f(x)=8x−2$

Part B: Gradients at Points

5. Find the gradient of $f(x)=2x^2−5$ at $x=3$.
6. For the curve $y=x^3−2x$, find the coordinates where the gradient is equal to 1.
7. Challenge: If $f(x)=ax^2+4x$ and $f′(1)=10$, find the value of $a$.

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Solutions (Summary)

• 1. $5x^4$
• 2. $20x−4$
• 3. $x^2$
• 4. 8
• 5. $f′(x)=4x$. At $x=3$, gradient = 12.
• 6. $\frac{dy}{dx}$​=$3x^2−2$. Set to 1: $3x^2−2$=1→$3x^2$=3→$x$=±1. Points are (1,−1) and (−1,1).
• 7. $f′(x)=2ax+4$. So, $2a(1)+4$=10→$2a$=6→a=3.