Product rule, quotient rule and the trigonometric derivatives

Learning objectives

Differentiate a product with (uv)′ = u′v + uv′, extending the pattern to a product of three factors

Differentiate a quotient with (u/v)′ = (u′v − uv′)/v², and judge when rewriting the quotient as a product with a negative exponent gives a shorter route

Derive the derivatives of tan x, cot x, sec x and csc x from the quotient rule applied to sine and cosine, and then recall them without re-deriving

Evaluate derivatives of products and quotients at a point from a table or a graph of f, g, f′ and g′ rather than from explicit formulas

Apply the product and quotient rules in context, for example to the rate of change of revenue as price times quantity, or of a concentration as a ratio of two changing amounts