More Rules of DIFFERENTIATION Finding Derivatives Differentiation Rules

More Rules of DIFFERENTIATION Finding Derivatives

Differentiation Rules REVIEW ü Constant Function Rule ü Power Rule ü Constant Multiple Rule ü Sum & Difference Rule

Derivatives of Products and Quotients • We will use a new Rule to differentiate products and quotients • Product Rule • Quotient Rule

Product Rule • The rule states: o o The derivative of the first function times the second function PLUS the derivative of the second function times the first function. How it looks…

Product Rule • EXAMPLE: o

Product Rule • EXAMPLE: o

Product Rule • EXAMPLE:

Product Rule • EXAMPLE:

Product Rule EXAMPLE: • Find the equation of the line tangent to the graph of f(x) at x=3 • Find the value of x where the tangent is horizontal.

Product Rule • EXAMPLE: o Find the equation of the line tangent to the graph of f(x) at x = 3 • The tangent line is y = -2 x – 18

Product Rule • EXAMPLE: o Find the value of x where the tangent is horizontal.

Product Rule • EXAMPLE: o One hour after a vampire has drained a victim, x milligrams of a particular virus is transmitted to the victim based on body temperature change. The change in body temperature T(x), in °F, is given by: o o Find T’(x) Find T’(1), T’(3) & T’(6).

Product Rule • EXAMPLE: o

Product Rule o EXAMPLE: Find T’(1), T’(3) & T’(6).

QUOTIENT Rule • The rule states: o o The derivative of the TOP function times the Bottom function MINUS the derivative of the Bottom function times the TOP function all divided by the Bottom function squared. How it looks…

QUOTIENT Rule • EXAMPLE: o

QUOTIENT Rule • EXAMPLE: o

QUOTIENT Rule • EXAMPLE: o

QUOTIENT Rule • EXAMPLE: o

QUOTIENT Rule • EXAMPLE: o Since Buffy came to Sunnydale, the total sales of wooden stakes t months after her arrival are given by: o Find S’(t) o Find S(10) and S’(10), then write an interpretation of the results. o Estimate the total sales after 11 months.

QUOTIENT Rule • EXAMPLE: o Since Buffy came to Sunnydale, the total sales of wooden stakes t months after her arrival are given by: o Find S’(t)

QUOTIENT Rule • EXAMPLE: o Since Buffy came to Sunnydale, the total sales (in $ dollars) of wooden stakes t months after her arrival are given by: o Find S(10) and S’(10), then write an interpretation of the results. o Sales after 10 months total $125. o Sales increase monthly at a rate of $18. 75.

QUOTIENT Rule • EXAMPLE: o Since Buffy came to Sunnydale, the total sales of wooden stakes t months after her arrival are given by: o Estimate the total sales after 11 months. o Sales after 11 months are approximately $143. 75.

Quotient Rule EXAMPLE: According to economic theory, the demand x for a quantity in a free market decreases as the price p increases. Suppose that the number x of HDTVs people are willing to buy per week from a retail chain at a price of $p is given by: • Find and interpret • Estimate demand when the price is $41. when p = $40.

Quotient Rule EXAMPLE: According to economic theory, the demand x for a quantity in a free market decreases as the price p increases. Suppose that the number x of HDTVs people are willing to buy per week from a retail chain at a price of $p is given by:

Quotient Rule EXAMPLE: According to economic theory, the demand x for a quantity in a free market decreases as the price p increases. Suppose that the number x of HDTVs people are willing to buy per week from a retail chain at a price of $p is given by: At $40, the demand for HDTVs is 800 in weekly sales. However, at the same price level, demand is decreasing at a rate of 16 per week.

Quotient Rule EXAMPLE: According to economic theory, the demand x for a quantity in a free market decreases as the price p increases. Suppose that the number x of HDTVs people are willing to buy per week from a retail chain at a price of $p is given by: At $41, the demand for HDTVs is 784 sales.

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