Business Calculus

More Derivative Applications

2.6 Differentials

We often use Δx to indicate a small change in x, and Δy for asmall change in y. It is important to note that Δy refers to achange in the height of the actual function.

When we take a derivative, we find the slope of the tangent line,dy/dx. We can relate this to slope in terms of rise over run.

When looking at the tangent line:Rise = dyRun = dx

dx and dy are called differentials.

In applications, we are given the amount of change in x, and areasked to find the resulting change in y.

If we are looking for the exact change in y, we find Δy:

If we want to approximate the change in y, we can use the tangent line instead of the curve itself. In other words, approximate change in y is dy:

dy is a reasonable approximation to Δy as long as Δx is ‘small’.

)()( xfxxfy

dxxfdy )(

In business, we are interested in how a small change in the number of items produced and sold can affect cost, revenue,and profit.If we are given Cost, Revenue, and Profit functions where theinput is x number of items, then:

Marginal Cost is the approximate additional cost to produce the next item.

Marginal Revenue is the approximate additional revenue earned inselling the next item.

Marginal Profit is the approximate additional profit when producing and selling the next item.

Marginal Analysis

If we are finding the approximate change in cost, revenue, orprofit for one additional item, we set Δx = 1. This means that, for a given number of items, x, produced and sold:

The cost to produce the x + 1st item is: Marginal Cost = C′(x)

The revenue earned for selling the x + 1st item is: Marginal Revenue = R′(x)

The profit earned for the x + 1st item is: Marginal Profit = P′(x)

2.7 Implicit Differentiation

When using the chain rule to find a derivative, we look for an‘inside’ function and ‘outside’ function.

example:

outside =

inside =

The chain rule says:

The derivative of f is:

72 )43()( xxf

7) (

43 2 x

43) ( 27 xdx

d

dx

d

xx 6)43(7 62

Now consider the possibility that we are told the ‘outside’ function,but we do not know the ‘inside’ function.We would need a ‘place holder’ for the inside function. We will use a variable such as u or y to represent the ‘inside’ function.

example:

outside =

inside =

The chain rule says:

The derivative of f is:

7) (

u

udx

d

dx

d7) (

7) ()( uxf

udx

du 6) (7

Since we are not told the actual formula for the ‘inside’ function,we will write:

When given an equation with two or more letters, we will identify one letter as the input variable, and one letter as theoutput variable. All other letters are considered constants orcoefficients.

Our goal is to find a particular derivative, such as , whengiven an equation involving the two variables.

dx

duuxf 67)(

Implicit Differentiation

dx

dy

Important Fact: Since the equation is not stated as a normalfunction, i.e. y = f (x), we must assume that the output variablein the equation represents a function that is currently unseen.Therefore, we will think of the output variable as a hidden ‘inside’function.

To perform implicit differentiation:1. Identify the input and output variables.2. Take the derivative of both sides of the equation with respect

to the input variable. Remember that when the outputvariable is involved in the differentiation, it needs to beregarded as an ‘inside’ function.

3. Solve for the derivative symbol (such as ). dx

dy

2.7 Related Rates

In some application problems, we are interested in how cost, revenue, or profit might change with respect to time.

We will be given information about cost, revenue, and profit asfunctions of x, which represents the number of items produced andsold. We are also given information about how x, the number of items produced and sold, will change over time.Our goal is to determine how the passing of time will affect the cost, revenue, and profit.

Since a derivative is an instantaneous rate of change, we willuse the derivative symbol to represent these changes.

represents the change in number of items with respect to time.

represents the change in revenue with respect to number of items sold.

represents the change in revenue with respect to time.

is the relationship which allows us to find how a small change in time will cause a change in revenue.

dt

dx

dt

dR

dx

dR

dt

dx

dx

dR

dt

dR