Differentiation and Calculus Fundamentals (HSC 2026)

A comprehensive set of vocabulary flashcards covering the essential formulas, rules, and conceptual conditions of differentiation for the HSC 2026 syllabus.

Basic Sine Limit

$$\lim_{x \to 0} \frac{\sin(x)}{x} = 1$$

Exponential Limit (Base e)

$$\lim_{x \to 0} \frac{e^x - 1}{x} = 1$$

Natural Logarithm Limit

$$\lim_{x \to 0} \frac{\ln(1 + x)}{x} = 1$$

Power Rule Limit

$$\lim_{x \to a} \frac{x^n - a^n}{x - a} = n a^{n - 1}$$

Compound Interest Limit Form

$\lim_{x \to 0} (1 + x)^{\frac{1}{x}} = e$ or $\lim_{x \to \infty} \left(1 + \frac{1}{x}\right)^x = e$

Limit of (a^x - 1)/x

$$\lim_{x \to 0} \frac{a^x - 1}{x} = \ln(a)$$

Expansion of e^x

$$1 + \frac{x}{1!} + \frac{x^2}{2!} + \frac{x^3}{3!} + \dots$$

Expansion of sin(x)

$$x - \frac{x^3}{3!} + \frac{x^5}{5!} - \dots$$

First Principles of Differentiation

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

Derivative of a Constant

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

Derivative of sin(ax)

$$a\cos(ax)$$

Derivative of cos(ax)

$$-a\sin(ax)$$

Derivative of tan(x)

$$\sec^2(x)$$

Derivative of sec(x)

$$\sec(x)\tan(x)$$

Product Rule

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

Quotient Rule

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

Derivative of sin^-1(x)

$$\frac{1}{\sqrt{1 - x^2}}$$

Derivative of cos^-1(x)

$$-\frac{1}{\sqrt{1 - x^2}}$$

Derivative of tan^-1(x)

$$\frac{1}{1 + x^2}$$

Derivative of log_a(x)

$$\frac{1}{x}\log_a(e)$$

Derivative of a^x

$$a^x\ln(a)$$

Successive Differentiation Symbol

$y_n$, representing the nth derivative relative to x.

Slope of a Tangent (m)

Evaluated as $m = \left(\frac{dy}{dx}\right)_{(x_1, y_1)}$ at the point $(x_1, y_1)$.

Slope of a Normal

The negative reciprocal of the tangent slope, given by $-\frac{1}{m}$.

Increasing Function Condition

A function $y = f(x)$ is increasing on an interval if $f'(x) > 0$ for all x in that interval.

Decreasing Function Condition

A function $y = f(x)$ is decreasing on an interval if $f'(x) < 0$ for all x in that interval.

Maximum Value (Maxima) Condition

Requires $f'(x) = 0$ and the second derivative $f''(x) < 0$.

Minimum Value (Minima) Condition

Requires $f'(x) = 0$ and the second derivative $f''(x) > 0$.

Inflection Point

A point on a curve where the second derivative $f''(x) = 0$.

Implicit Function

A function where the dependent variable is not isolated on one side, such as $x^3 + x^2y + xy^2 = 0$.