Transcript G. Derivatives of Transcendental Functions

G. Derivatives of
Transcendental Functions
CALCULUS 30
1. Differentiating Logarithmic Functions
 Logarithmic functions were introduced in Math
B30.
 We will review basic properties of logarithms, and
then move in differentiating them.
 Laws of Logarithms
1. Product Law
2. Quotient Law
3. Power Law
 There are 2 special types of logarithms we could see
1. Common Logarithm – has base 10 (which is usually
not written) → logx
2. Natural Logarithm – has base e
(e =2.71828182845...) →lnx
Example
 Write each of the following as the logarithm of a
single term. You do not have to evaluate the
logarithm.
 We now move on to the derivative of logarithmic
functions.
Examples
 Find the derivative of
 What if there is a function inside the logarithm? Our
derivative changes slightly.
Examples
 Find the derivative of
 Sometimes, depending on the functions given inside
the logarithm, it may be easier to use the law of
logarithms first, before trying to differentiate.
Examples
 Find the derivative of the following by using
logarithm laws before differentiating
 Finally, we may also be asked to use more complex
methods to differentiate functions involving
logarithms.
Example
 Find the derivative of the function
Assignment
 Ex. 7.1 (p. 303) #1-9 odds in each
2. Differentiating Exponential Functions
 An exponential function is something that looks like
𝑦 = 𝑏𝑥
 You may think (based on the power rule) that the
derivative of 𝑦 = 𝑏 𝑥 is
𝑑𝑦
𝑑𝑥
 This is NOT the case.
= 𝑥𝑏 𝑥−1
 There are 3 situations that you will encounter when
dealing with exponential functions.
Example
Examples
Examples
Assignment
 Ex. 7.2 (p. 311) #1-45 odds
3. Limits Involving Trigonometric Functions
 In our study of limits, we did not touch trigonometric
functions.
 This is because there are special properties when we
look at limits involving trigonometric functions.
 They are:
 In any of the questions you will be asked, you will
need to manipulate your expression into looking like
one of these before evaluating the limit.
Example
Example
Example
Example
Assignment
 Ex. 7.3 (p. 317) #1-28 odds
4. Derivatives of Sine and Cosine
 There are 2 situations for each function.
 One in which there is just an x inside the function,
and another where there is some other function
inside the trigonometric function.
Examples
 Find the derivative of each of the following functions.
𝐼𝑓 𝑦 = cos 𝑥 ,
𝐼𝑓 𝑦 = cos 𝑢 ,
𝑑𝑦
𝑡ℎ𝑒𝑛
= −𝑠𝑖𝑛𝑥
𝑑𝑥
𝑑𝑦
𝑑𝑢
𝑡ℎ𝑒𝑛
=−
sin 𝑢
𝑑𝑥
𝑑𝑥
Example
 Find the derivative of each function
Assignment
 Ex. 7.4 (p. 324) #1-57 odds