Year 12 Calculus Reference Guide Flashcards

A set of vocabulary flashcards covering core limits, differentiation rules, advanced calculus theorems, and basic integration formulas based on the Year 12 Calculus Reference Guide.

Derivative from First Principles

The formal definition defined as $f'(x) = \text{lim}_{h \rightarrow 0} \frac{f(x + h) - f(x)}{h}$

Constant Rule (Differentiation)

The derivative of a constant $c$ is $0$.

Power Rule (Differentiation)

For a function $x^n$, the derivative is $n \times x^{n-1}$.

Constant Multiple Rule (Differentiation)

The derivative of $c \times f(x)$ is $c \times f'(x)$.

Sum / Difference Rule (Differentiation)

The derivative of $f(x) \text{ ± } g(x)$ is $f'(x) \text{ ± } g'(x)$.

Exponential Derivative Rule

The derivative of $e^x$ is $e^x$.

Natural Log Derivative Rule

The derivative of $\text{ln}(x)$ is $\frac{1}{x}$.

Sine Function Derivative Rule

The derivative of $\text{sin}(x)$ is $\text{cos}(x)$.

Cosine Function Derivative Rule

The derivative of $\text{cos}(x)$ is $-\text{sin}(x)$.

Product Rule

Used when two functions are multiplied together: $\frac{d}{dx}[u \times v] = u \times \frac{dv}{dx} + v \times \frac{du}{dx}$

Quotient Rule

Used when one function is divided by another: $\frac{d}{dx}[\frac{u}{v}] = \frac{v \times \frac{du}{dx} - u \times \frac{dv}{dx}}{v^2}$

Chain Rule

Used for composite functions: $\frac{dy}{dx} = \frac{dy}{du} \times \frac{du}{dx}$

Integration

A process that reverses differentiation to find the original function or the total area under a curve.

Constant of Integration (C)

A value that must always be added to the result of an indefinite integral.

Indefinite Integral of a Constant (k)

\text{∫} k \text{ } dx = k \times x + C

Power Rule for Integration

For $x^n$ where $n \neq -1$, the integral is $\frac{x^{n+1}}{n + 1} + C$

Indefinite Integral of 1/x

\text{∫} x^{-1} \text{ } dx = \text{ln}|x| + C

Indefinite Integral of Exponential Function

\text{∫} e^x \text{ } dx = e^x + C

Indefinite Integral of Cosine Function

\text{∫} \text{cos}(x) \text{ } dx = \text{sin}(x) + C

Indefinite Integral of Sine Function

\text{∫} \text{sin}(x) \text{ } dx = -\text{cos}(x) + C