Calculating the Derivative Chapter 12

Ch. 12 Calculating the Derivative 12.1 Techniques for Finding Derivatives 12.2 Derivatives of Products and Quotients 12.3 The Chain Rule 12.4 Derivatives of Exponential Functions 12.5 Derivatives of Logarithmic Functions

12.1 Techniques for Finding the Derivative Using the definition (below) to calculate the derivative can be very involved. Section 11.5 introduces rules to simplify the calculation of derivatives. The simplified process does not change the interpretation of the derivative.

Techniques for Finding the Derivative NOTATIONS FOR THE DERIVATIVE The derivative of y = f(x) may be written in any of the following ways: Each of the above represents the “derivative of the function f(x) (or y) with respect to x.”

CONSTANT RULE If f(x) = k, where k is any real number, then f ‘(x) = 0 (The derivative of a constant is 0) If f(x) = 12, then f ‘(x) = 0 If p(t) = , then Dt[p(t)] = 0 If y = 23, then dy/dx = 0

POWER RULE If f(x) = xn for any real number n, then f ‘(x) = nxn – 1 (The derivative of f(x) = xn is found by multiplying by the exponent n and decreasing the exponent on x by 1.) If y = x3, then Dxy = 3x3 – 1 = 3x2 If y = x, then dy/dx = 1x1 – 1 = x0 = 1

CONSTANT TIMES A FUNCTION Let k be a real number. If f ‘(x) exists, then Dx[kf(x)] = kf ‘(x) (The derivative of a constant times a function is the constant times the derivative of the function.)

A FUNCTION DIVIDED BY A CONSTANT Let k be a real number. If f ‘(x) exists, then (The derivative of a function divided by a constant is the derivative of the function divided by the constant.)

SUM OR DIFFERENCE RULE If f(x) = u(x) ± v(x) , and if u’(x) and v’(x) exist, then f ‘(x) = u’(x) ± v’(x) (The derivative of a sum or difference of functions is the sum or difference of the derivatives.) y = 2x5 + 6x3 let u(x) = 2x5 and v(x) = 6x3, then y = u(x) + v(x) Since u’(x) = 10x4 and v’(x) = 18x2, dy/dx = 10x4 + 18x2 f(x) = (4x2 – 3x)2 f(x) = 16x4 – 24x3 + 9x2 [(a – b)2 = a2 – 2ab + b2] f ‘x = 64x3 – 72x2 + 18x

Marginal Analysis Marginal Revenue (MR) is the derivative of the Total Revenue function. When MR = 0, revenue is maximized! Marginal Cost (MC) is the derivative of the Total Cost function. When MC = 0, cost is minimized! When MR = MC, profit is maximized! Marginal Product (MP) is the derivative of the Total Product function. Marginal Propensity to Consume (MPC) is the derivative of the consumption function.

Marginal Cost – example A firm’s total cost in hundreds of dollars to produce x thousand widgets is given by C(x) = 0.1x2 + 3 Find the marginal cost of producing at x = 10,000 widgets. MC = C’(x) C’(x) = 0.2x Therefore, the marginal cost at x = 10,000 widgets is MC = 0.2(10) = 2 or $200 After 10,000 widgets have been produced, the cost to produce 1,000 more widgets will be approximately $200

Marginal Cost – another example Average Cost (AC) = TC/Output If a manufacturer’s average cost equation is AC = 0.0001q2 - 0.02q + 5 + (5000 / q, where q = output Find the marginal cost function Calculate MC when q = 50

Since AC = TC/q, then TC = q(AC) Differentiating TC gives MC MC (at q = 50) = 0.0003(50)2 – 0.04(50) + 5 = $3.75

Now You Try Assume that a demand equation is given by Find the marginal revenue at an output of 1000 units. Hint: Solve the demand equation for p and use R(x) = xp

12.2 Derivatives of Products and Quotients PRODUCT RULE If f(x) = u(x) • v(x), and u’(x) and v’(x) both exist, then f ’(x) = u(x) • v’(x) + v(x) • u’(x) (The derivative of a product of two functions is the first function times the derivative of the second, plus the second function times the derivative of the first.) Consider f as the product of two functions: (x2 + 3x) = h(x) and (4x + 5) = g(x), so, f(x) = h(x)· g(x).

f ’(x) = h(x)g’(x) + g(x)h’(x) PRODUCT RULE apply the product rule, f ’(x) = h(x)g’(x) + g(x)h’(x) f ’(x) = (x2 + 3x) (4) + (4x + 5) (2x + 3) f ’(x) = 4x2 + 12x +8x2 + 12x + 10x + 15 = 12x2 + 34x + 15 f(x) = h(x)· g(x).

Derivatives of Products and Quotients QUOTIENT RULE If f(x) = u(x)/v(x), and all indicated derivatives exist, and v(x)  0, then (The derivative of a quotient is the denominator times the derivative of the numerator, minus the numerator times the derivative of the denominator, all divided by the square of the denominator.)

QUOTIENT RULE Example: Find f ’(x) if:

Quotient Rule and Marginal Revenue Suppose that a firm’s total revenue function is R = 1000q / (q + 5). Find the marginal revenue function. How much additional revenue will the 46th unit sold bring in? (Calculate MR at q = 45)

Now You Try. A company that manufactures bicycles has determined that a new employee can assemble M(d) bicycles per day after d days of on-the-job training, where Find the rate of change function for the number of bicycles assembled with respect to time. Find and interpret M’(2) and M’(5).

12.3 The Chain Rule Composite Functions S = f(I) I = g(t) S = f [g(t)] The amount of money Sharon saves (S) depends upon her income (I). We can express this as S = f(I) Sharon’s income increases each year (t) she continues to work. So, I = g(t) The composition of functions f and g: S = f [g(t)]

Composite Functions - example

The Chain Rule Used to find the derivative of a composite function. If y is a function of u, or y = f(u), and if u is a function of x, or u = g(x), then y = f(u) = f[g(x)], and

The Chain Rule – example Think of y = (x2 + 1)3 as a composite of two functions: y = u3 and u = x2 + 1 Then, Replace u with x2 + 1

Now You Try Assume that the total revenue from the sale of x television sets is given by Find the marginal revenue when the following numbers of sets are sold 400 500 600

12.4 Derivatives of Exponential Functions This section examines four exponential functions: Derivative of ex Derivative of ax Derivative of ag(x) Derivative of eg(x)

Derivatives of Exponential Functions DERIVATIVE OF ex Dx[ex] = ex DERIVATIVE OF ax Dx[ax] = (ln a)ax y = 5t

Derivatives of Exponential Functions DERIVATIVE OF ag(x) Dx[ag(x)] = (ln a) ag(x)g’(x) Let g(x) = 5x, with g’(x) = 5. Then,

Derivatives of Exponential Functions DERIVATIVE OF eg(x) Dx[eg(x)] = eg(x)g’(x) Let g(x) = 2x, with g’(x) = 2. Then,

Example The sales of new personal computers (in thousands) are given by S(t) = 100 – 90e- 0.3t where t represents time in years. Find the rate of change of sales After 1 year After 5 years

Example 27e -0.3(1) = 20 (or 20,000) 27e -0.3(5) = 6 (or 6,000) Let g(x) = -0.3t, with g’(x) = -0.3. Then, 27e -0.3(1) = 20 (or 20,000) 27e -0.3(5) = 6 (or 6,000)

12.5 Derivatives of Logarithmic Functions DERIVATIVE OF ln |x|

Derivatives of Logarithmic Functions DERIVATIVE OF [ln |g(x)|]

Now You Try Suppose the demand function for x thousand units of a certain item is where p is in dollars. Find the marginal revenue if quantity demanded is 8000 units.

 Chapter 12 End