G. Derivatives of Transcendental Functions
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Transcript G. Derivatives of Transcendental Functions
G. Derivatives of
Transcendental Functions
CALCULUS 30
C30.7
DEMONSTRATE UNDERSTANDING OF
TRANSCENDENTAL FUNCTION DERIVATIVES
AND THEIR APPLICATIONS.
1. Differentiating Logarithmic Functions
C30.7
Demonstrate understanding of transcendental
function derivatives and their applications.
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
C30.7
Demonstrate understanding of transcendental
function derivatives and their applications.
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
C30.7
Demonstrate understanding of transcendental
function derivatives and their applications.
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
C30.7
Demonstrate understanding of transcendental
function derivatives and their applications.
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