Chapter 5-Derivatives of Exponential and Trigonometric Functions

By Jeffrey Kim, Chris Bayley, Jacqueline Tennant, Fredy Valderrama

Agenda

• Review of Pre-requisite Skills• Derivative of General Exponential Functions• Derivative of the Exponential Function• Optimization Problems with Exponential

Functions• Derivatives of Sinusoidal Functions

Things you should know:Exponent Laws

Rules Initial Final

Product (am)(an) am+n

Quotient (am)/(an) am-n

Power (am)n amn

Negative Exponents a-n 1/an

Things you should know:Logarithm Laws

Rules Initial Final

Product Logb m + Logb n LogbmnQuotient Logb m – Logb n Logb(m/n)Power Logbmn n Logb mBase Logbbm m

Evaluating without a standard base

Logb m Log m Log b

More Things You Should Know1. Product rule ()=f()()

’()=f’()()+f()’()

2. Quotient rule ()= ’()=

3. Chain rule ()= f(())’()=f’(()) ’()

’()=

f()=6

f()=f’()=

f’()=

()= ’()=(’()=)()+

f’()=6(6-5)f’()=(36-30)

f(x) = ex f’(x) = ex

g(x) = eh(x) g’(x) = eh(x) * h’(x)

K/U question

Differentiate the following function:

5.2: Derivative of the General Exponential Function, y = bx

Consider the function f(x) = 3x.

x f(x) f’(x) f’(x)/f(x)

-2 1/9 0.1220681 1.10

-1 1/3 0.3662042 1.10

0 1 1.0986125 1.10

1 3 3.2958375 1.10

2 9 9.8875126 1.10

3 27 29.662538 1.10

Key Points:

• f’(x) is a vertical stretch or compression of f(x), dependent on the value of b

• the ratio f’(x)/f(x) is a constant and is equal to the stretch/compression factor

Derivative of f(x) = bx:f’(x) = (ln b) * bx

Derivative of f(x) = bg(x):f’(x) = bg(x) * (ln b) * g’(x)

Graphing f(x) and f’(x):

f’(x) = 1.10 • 3x

f(x) = 3x

Examples

Ex 1 find the derivative of f(x)a) f(x) = 8x

b) f(x) = 34+2

Knowledge 1) Find the derivative of f(x)

a) f(x) = 8x

b) f(x) = 5 2x^2 – 3x + 10

Question for Test

Question for Test Application

You purchased a new car for $16,000. the value of the car after t years is given by the function, V(t) where t is the number of years after the purchased and v(t) is the value of the car in dollars

V(t)=16000(0.78)t

a) Determine the value of the car after the first year.b) Find the rate of change when t=1c) Interpret the results.

5.3Optimization Problems with Exponential Functions

Algorithm for Solving Optimization Problems:

1. Understand the problem, and identify quantities that can vary. Determine a function in one variable that represents the quantity to be optimized.

2. Whenever possible, draw a diagram, labelling the given and required quantities.

3. Determine the domain of the function to be optimized, using the information given in the problem.

4. Use the algorithm for extreme values to find the absolute maximum or minimum value in the domain.

5. Use your result for step 4 to answer the original problem.

Application Question

Jack, given $24 from his parents, wants to bake cookies for profit. The cost of baking 1 batch of 24 cookies is $12. If his revenues from the sale of these cookies are modeled by f(x) = e3x – 150x, and Jack must sell all his cookies after baking before baking more cookies, find the number of cookies Jack must sell after his initial round of baking that maximizes his profits.

Derivatives of Sinusoidal Functions

Key Concepts

• F(x)=sinx , then = cosx• F(x)=cosx, then =-sinx

Composite sinusoidal functions• If y=sinf(x), then y’ = cosf(x)× f’(x) • If y=cosf(x) , then = -sinf(x) × f’(x)

Knowledge Question

Find the derivative of y = tan(x)

Homework to review