Last updated on Sep 21, 2026

First Principles for Derivatives (HSC SSCE Mathematics Advanced): Quizzes

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First Principles for Derivatives

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What does the derivative of a function represent geometrically?

Gradient of tangent line at a point

What is the first principles formula for finding $f'(x)$ ?

$\lim_{h \to 0} \frac{f(x+h) - f(x)}{h}$

What does $h \to 0$ mean in the first principles formula?

As $h$ gets infinitely close to zero

What is the gradient of the secant line through $P(x, f(x))$ and $Q(x+h, f(x+h))$ ?

$\frac{f(x+h) - f(x)}{h}$

What must you do before cancelling $h$ in the first principles method?

Factorise $h$ from the numerator

What is the correct expansion of $(x+h)^2$ ?

$x^2 + 2xh + h^2$

Using first principles, what is the derivative of $f(x) = 3x + 2$ ?

$3$

What is $f'(x)$ for $f(x) = x^2 + 5x$ using first principles?

$2x + 5$

When should you substitute $h = 0$ in the first principles process?

After cancelling the common factor $h$

If $f'(x) = 4x - 3$ , what is the gradient at $x = 3$ ?

$9$

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First Principles for Derivatives (HSC SSCE Mathematics Advanced): Quizzes

Practise the questions

What does the derivative of a function represent geometrically?

What is the first principles formula for finding f′(x)f'(x)f′(x)?

What does h→0h \to 0h→0 mean in the first principles formula?

What is the gradient of the secant line through P(x,f(x))P(x, f(x))P(x,f(x)) and Q(x+h,f(x+h))Q(x+h, f(x+h))Q(x+h,f(x+h))?

What must you do before cancelling hhh in the first principles method?

What is the correct expansion of (x+h)2(x+h)^2(x+h)2?

Using first principles, what is the derivative of f(x)=3x+2f(x) = 3x + 2f(x)=3x+2?

What is f′(x)f'(x)f′(x) for f(x)=x2+5xf(x) = x^2 + 5xf(x)=x2+5x using first principles?

When should you substitute h=0h = 0h=0 in the first principles process?

If f′(x)=4x−3f'(x) = 4x - 3f′(x)=4x−3, what is the gradient at x=3x = 3x=3?

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Explore HSC SSCE Mathematics Advanced Model Answers by Topics

Algebraic Techniques

Solve Equations

Introduction to Functions and Relations

Linear Functions

Quadratic and Cubic Functions

Function Properties

Functions and Relations

Trigonometry

Trigonometric Identities and Equations

Probability

Data

Introduction to Rates of Change

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Exponential Functions

Logarithmic Functions

Transformations

Trigonometric Identities and Equations

Probability

Data

Introduction to Rates of Change

The Derivative

Exponential Functions

Logarithmic Functions

Transformations

Advanced Applications of Transformations

Explore HSC SSCE Mathematics Advanced Flashcards by Topics

Algebraic Techniques

Solve Equations

Introduction to Functions and Relations

Linear Functions

Quadratic and Cubic Functions

Function Properties

Functions and Relations

Trigonometry

Trigonometric Identities and Equations

Probability

Data

Introduction to Rates of Change

The Derivative

Exponential Functions

Logarithmic Functions

What is the first principles formula for finding $f'(x)$ ?

What does $h \to 0$ mean in the first principles formula?

What is the gradient of the secant line through $P(x, f(x))$ and $Q(x+h, f(x+h))$ ?

What must you do before cancelling $h$ in the first principles method?

What is the correct expansion of $(x+h)^2$ ?

Using first principles, what is the derivative of $f(x) = 3x + 2$ ?

What is $f'(x)$ for $f(x) = x^2 + 5x$ using first principles?

When should you substitute $h = 0$ in the first principles process?

If $f'(x) = 4x - 3$ , what is the gradient at $x = 3$ ?