Grade 12

Mathematics

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Online Classroom Rules

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Apply the Product Rule to find derivatives

Learning Objectives

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Apply the Quotient Rule to find derivatives

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Solve real-life problems using the Product and Quotient Rules

Given our experience with the product rule, you probably have no expectation that the derivative of a quotient will turn out to be the quotient of the derivatives.

Just to be sure, let’s try a simple experiment.

Solution Although it may be tempting to use the product rule for the first term and

the quotient rule for the second term, notice that it’s simpler to first rewrite the function.

We can combine the two powers of x in the first term. Since the second term is a

fraction with a constant numerator, we can more simply write it using a negative

exponent. We have

(e.g., if you sell 10 items at $4 each, you earn $40). Since these quantities are changing in time,

we write R(t) = Q(t)P(t), where R(t) is revenue, Q(t) is quantity sold and P(t) is the price, all at time t.

We don’t have formulas for any of these functions, but from the product rule, we have

We have information about each of these terms:

the initial price, P(0), is 25 (dollars); the rate of change of the price is P’(0) = 2 (dollars per year);

the initial quantity, Q(0), is 150 (thousand items) and the rate of change of quantity is Q’(0) = −8 (thousand items per year). Note that the negative sign of Q’(0) denotes a decrease in Q.

Thus, R’(0) = (−8)(25) + (150)(2) = 100 thousand dollars per year. Since the rate of change is positive, the revenue is increasing.

To answer these questions, we need the basic relationship

revenue = quantity × price

Both the numerator and denominator are positive, so u(m) > 0.

A positive slope for all tangent lines indicates that the graph of u(m) should rise from left to right. (See Figure 2.23.) Said a different way, u(m) increases as m increases.

In golf terms, this says that (all other things being equal) the greater the mass of the club, the greater the velocity of the ball will be.

Finally, we compute u(0.15) = 103.75 and u(0.20) = 66.4.

This says that the rate of increase in ball speed is much less for the heavier club than for the lighter one. Since heavier clubs can be harder to control, the relatively small

increase in ball speed obtained by making the heavy club even heavier may not

compensate for the decrease in control.