# Hints

STACK contains a "formula sheet" of useful fragments which a teacher may wish to include in a consistent way. This is achieved through the "hints" system.

Hints can be included in any CASText.

To include a hint, use the syntax

```
[[facts:tag]]
```

The "tag" is chosen from the list below.

## All supported fact sheets

### The Greek Alphabet

`[[facts:greek_alphabet]]`

Upper case, | lower case, | name |

alpha | ||

beta | ||

gamma | ||

delta | ||

epsilon | ||

zeta | ||

eta | ||

theta | ||

kappa | ||

mu | ||

nu | ||

xi | ||

omicron | ||

pi | ||

iota | ||

rho | ||

sigma | ||

lambda | ||

tau | ||

upsilon | ||

phi | ||

chi | ||

psi | ||

omega |

### Inequalities

`[[facts:alg_inequalities]]`

### The Laws of Indices

`[[facts:alg_indices]]`

The following laws govern index manipulation:

### The Laws of Logarithms

`[[facts:alg_logarithms]]`

For any base with : The formula for a change of base is: Logarithms to base , denoted or alternatively are called natural logarithms. The letter represents the exponential constant which is approximately .

### The Quadratic Formula

`[[facts:alg_quadratic_formula]]`

If we have a quadratic equation of the form: then the solution(s) to that equation given by the quadratic formula are:

### Partial Fractions

`[[facts:alg_partial_fractions]]`

Proper fractions occur with when and are polynomials with the degree of less than the degree of . This this case, we proceed as follows: write in factored form,

- a
*linear factor*in the denominator produces a partial fraction of the form - a
*repeated linear factors*in the denominator produce partial fractions of the form - a
*quadratic factor*in the denominator produces a partial fraction of the form *Improper fractions*require an additional term which is a polynomial of degree where is the degree of the numerator (i.e. ) and is the degree of the denominator (i.e. ).

### Degrees and Radians

`[[facts:trig_degrees_radians]]`

### Standard Trigonometric Values

`[[facts:trig_standard_values]]`

### Standard Trigonometric Identities

`[[facts:trig_standard_identities]]`

### Hyperbolic Functions

`[[facts:hyp_functions]]`

Hyperbolic functions have similar properties to trigonometric functions but can be represented in exponential form as follows:

### Hyperbolic Identities

`[[facts:hyp_identities]]`

The similarity between the way hyperbolic and trigonometric functions behave is apparent when observing some basic hyperbolic identities:

### Inverse Hyperbolic Functions

`[[facts:hyp_inverse_functions]]`

### Standard Derivatives

`[[facts:calc_diff_standard_derivatives]]`

The following table displays the derivatives of some standard functions. It is useful to learn these standard derivatives as they are used frequently in calculus.

, constant | |

, any constant | |

### The Linearity Rule for Differentiation

`[[facts:calc_diff_linearity_rule]]`

### The Product Rule

`[[facts:calc_product_rule]]`

The following rule allows one to differentiate functions which are multiplied together. Assume that we wish to differentiate with respect to . or, using alternative notation,

### The Quotient Rule

`[[facts:calc_quotient_rule]]`

The quotient rule for differentiation states that for any two differentiable functions and ,

### The Chain Rule

`[[facts:calc_chain_rule]]`

The following rule allows one to find the derivative of a composition of two functions. Assume we have a function , then defining , the derivative with respect to is given by: Alternatively, we can write:

### Calculus rules

`[[facts:calc_rules]]`

**The Product Rule**

The following rule allows one to differentiate functions which are
multiplied together. Assume that we wish to differentiate with respect to .
or, using alternative notation,
**The Quotient Rule**

The quotient rule for differentiation states that for any two differentiable functions and ,
**The Chain Rule**

The following rule allows one to find the derivative of a composition of two functions.
Assume we have a function , then defining , the derivative with respect to is given by:
Alternatively, we can write:

### Standard Integrals

`[[facts:calc_int_standard_integrals]]`

cot | ||

( | ||

( | ||

### The Linearity Rule for Integration

`[[facts:calc_int_linearity_rule]]`

### Integration by Substitution

`[[facts:calc_int_methods_substitution]]`

### Integration by Parts

`[[facts:calc_int_methods_parts]]`

or alternatively:

### Integration by Parts

`[[facts:calc_int_methods_parts_indefinite]]`

or alternatively: