transcendental functions

Transce

ndental Functi

ons

N I CH O L E U

R I BE , A

LY S S A KA S H E R ,

J AS M I N

E CL A R K E - T

E L F E R , DU S T I N

NE S S

Please refer to the formula sheet throughout the slide-show.

DIFFERENTIATING LOG FUNCTIONSThe derivative of the logarithmic function y = ln x is

given by:

(lnx)=

For some problems, we can use the logarithmic laws to simplify our log expression before differentiating it.

EXAMPLE

Find the derivative of

y = ln 2x

ANSWERWe use the log law:

log ab = log a + log b

We can write our question as:

y = ln 2x = ln 2 + ln x

ANSWER CONT..Now, the derivative of a constant is 0, so

So we are left with

(from our formula above)

The final answer is…

𝑑𝑦𝑑𝑥

=1𝑥

We can see from the following graph that

the slope of y = ln 2x (curve in black,

tangent in red) is the same as the slope of

y = ln x (curve in blue, tangent in pink), at

the point x = 2

THE UUnfortunately, we can only use the logarithm laws to help us in

a limited number of logarithm differentiation question types.

Most often, we need to find the derivative of a logarithm of some function of x. For example, we may need to find the

derivative of y = 2 ln ( − 1).

We need the following formula to solve such problems:

ANSWERWe put

u = − 1

Then the derivative of u is given by:u' = 6xSo the final answer is…

DERIVATIVES OF EXPONENTIAL FUNCTIONSThe derivative of ex is quite remarkable. The expression for the

derivative is the same as the expression that we started with; that is, ex!

What does this mean? It means the slope is the same as the function value (the y-value) for all points on the graph

SO WHATS THE DERIVATIVE OF Y=3409850934580349853458903458390458345838450I834792387494939423943493789E^X?

TRICK QUESTION!The answer is…

Crazy huh?

ITS U AGAIN?If u is a function of x, we can obtain the derivative of an expression in

the form :

EXAMPLE

DAT LOG FUNCTION YOWe can obtain the derivative of the logarithm function with

base b using:

We can write our question as:We begin by using the following log rule to

simplify our question:

𝒚=𝐥𝐨𝐠𝟐𝟔 𝒙=𝐥𝐨𝐠𝟐𝟔+𝐥𝐨𝐠𝟐 𝒙

THE FIRST TERM, LOG26, IS A CONSTANT, SO ITS DERIVATIVE IS 0. THE DERIVATIVE OF THE SECOND TERM IS AS FOLLOWS, USING OUR FORMULA:

T h e t e r m on t h e t o

p , lo g 2

e , is a

c o n s t an t . I

f we n

e e d a d

e c i ma l v

a l ue , w

e

c a n wo r k i t

ou t u

s i ng c

h a n g e of

b a s e

a s f ol l o

w s :

INTEGRATION

T R A N S C E N D E N T A L FU N C T I O

N S

INTEGRATION OF LOGSThe general power formula that we saw in Section 1 is

valid for all values of n except n = -1. =(

We take the opposite of the derivative of the logarithmic function to solve such cases:

The (absolute value) signs around the u are necessary since the log of a negative number is not defined.

EXAMPLE

ANSWER

K=C=Some Constant

SO, TOUGH GUY. WHAT'S

THE ANSWER TO THIS

QUESTION?

?

YEAH, YEAH, YEAH.

Whatever, you cheated and looked at the formula sheet.

ONE LAST THING..Whats the answer to this?

Yeah, you knew it the whole time.

WELP.That was transcendental functions.

Minus all the trigonometry junk awesomeness.

YOU’RE FINISHED!

CONGRATULATIO

NS!Yo

u’re now a

transce

ndental

ist!