The Product and Quotient Rules, and Higher Derivatives

The Product Rule is a formula used to find the derivative of a function that is the product of two differentiable functions. If you have a function h(x) = f(x) * g(x), where f and g are differentiable, the Product Rule states:

h'(x) = f'(x) * g(x) + f(x) * g'(x)

In words: the derivative of the first function times the second function, plus the first function times the derivative of the second function.

The Quotient Rule is a formula used to find the derivative of a function that is the ratio (quotient) of two differentiable functions. If you have a function h(x) = f(x) ÷ g(x), where f and g are differentiable and g(x) ≠ 0, the Quotient Rule states:

h'(x) = [f'(x) * g(x) - f(x) * g'(x)] ÷ [g(x)]²

A common mnemonic is "Low dHigh minus High dLow, over the square of the Low" (where 'Low' is g(x), 'High' is f(x), and 'd' means derivative).

Higher order derivatives are the derivatives of derivatives. If you take the derivative of a function f(x), you get the first derivative, denoted f'(x). If you then take the derivative of f'(x), you get the second derivative, denoted f''(x) or d2y ÷ dx2. Taking the derivative of the second derivative gives the third derivative, f'''(x) or d3y ÷ dx3, and so on for any positive integer order 'n', denoted f⁽ⁿ⁾(x).

The first derivative tells you about the rate of change and slope. The second derivative tells you about the rate of change of the slope (concavity). Higher derivatives provide further information about the function's behavior.

To find higher order derivatives of functions that involve products or quotients, you apply the Product Rule or Quotient Rule repeatedly.

For example, to find the second derivative of h(x) = f(x) * g(x):

1. First, use the Product Rule to find the first derivative, h'(x) = f'(x)g(x) + f(x)g'(x).
2. Then, take the derivative of h'(x) to find the second derivative, h''(x). You will need to apply the Product Rule *again* to each term in h'(x) (specifically to f'(x)g(x) and f(x)g'(x)) and then use the Sum Rule.

The process for the Quotient Rule is similar but often involves more complex algebra due to the denominator term and its square.