Differentiation Rules for Products and Quotients

Product Rule

A formula used to find the derivative of the product of two differentiable functions.

Quotient Rule

A formula used to find the derivative of a quotient of two differentiable functions.

f(x)g(x)

The product of two functions f(x) and g(x), where the derivative is found using the Product Rule.

Lo d-Hi

A mnemonic for remembering the Quotient Rule: Low times the derivative of High minus High times the derivative of Low, all over Low squared.

(uv)' = uv' + vu'

The formula for the derivative of a product of two functions u and v.

g(x)f'(x) - f(x)g'(x)

The formula for the derivative of a quotient of two functions f and g.

Differentiable

A function is differentiable at a point if it has a derivative at that point.

Higher-Order Derivative

The derivative of a function's derivative, e.g., second derivative f''(x).

$y = f(x)g(x)$

An expression defining y as the product of functions f(x) and g(x).

Critical Points

Points where the derivative of a function is zero or undefined, indicating potential maxima or minima.

Chain Rule

A rule for computing the derivative of the composition of two or more functions.

$f'(x)$

The first derivative of the function f(x).

$y' = f(x)g'(x) + g(x)f'(x)$

The formula resulting from applying the Product Rule to differentiate y = f(x)g(x).

$f(x) = (3x^2 - 1)(x^2 + 5x)$

An example function for practicing the Product Rule.

Polynomials

Mathematical expressions involving a sum of powers in one or more variables multiplied by coefficients.

Non-commutativity

A property of operations where changing the order of the operands changes the result, as in subtraction.

$u = 3x^2 - 1$

One part of the function used in the example for the Product Rule.

$v = x^2 + 5x$

The other part of the function used in the example for the Product Rule.

Complex Combinations

Functions that involve products, quotients, or compositions of multiple basic functions.

Rate of Change

A measure of how much a quantity changes in relation to the change in another quantity.

Simplification

The process of reducing a mathematical expression to its simplest form.

Product of Changes

The assumption that the derivative of a product is equal to the product of the derivatives, which is incorrect.

Explicit Derivative Calculation

The detailed process of applying differentiation rules step by step to find a derivative.

Constant Denominator

A denominator that does not contain any variables and can simplify derivative calculations.

Second Derivative Notation

Denoted as $f''(x)$, representing the derivative of the first derivative.

$y = rac{5x - 2}{x^2 + 1}$

An example function for practicing the Quotient Rule.

Correct Order in Quotient

Using the correct sequence (Lo d-Hi) in applying the Quotient Rule to avoid sign errors.

Parentheses in Derivatives

Using parentheses appropriately to clarify operations, especially with negative signs in rules.