Differentiation rule - ch 3

Question 1

What is the normal line?

Answer 1

perpendicular to tangent line

slope = negative reciprocal

Question 2

Is this true

Derivative of a constant times a function is the constant times derivative of the function?

Answer 2

True.

Constant pops out.

Question 3

What’s the definition and significance of the number e?

Answer 3

e^x is its own derivative.
Slope of y=e^x is equal to its y coordinate of the point (p 179)

Question 4

What type of real-world applications are trig functions used for?

Answer 4

quantities that vary in periodic manner

e.g.
vibrations, waves, elastic motions

Question 5

What’s the product rule of derivatives?

Answer 5

Question 6

What’s the quotient rule of differentiation?

Answer 6

Question 7

Table of differentiation formulas to know

Answer 7

Question 8

Derivatives of trig functions

Answer 8

The trig functions that start with letter “c” have negative derivative

Question 9

what is derivative of a^x

Answer 9

(a^x) (ln a)

So f’(e^x) = e^x.
= repeat * ln base

Question 10

Is e^x the only function that is its own derivative?

Answer 10

No

y=0

Question 11

lim (x –> 0) sin(h) / h =

Answer 11

1

Question 12

What is the Chain Rule?

Answer 12

if g is differentiable at x and f is differentiable at g(x) then f(g(x)) is differentiable at x and …

F’(x) = f’(g(x))*g’(x)

dy/dx = dy/du * du/dx

Deriv outer of inner times deriv inner

Question 13

Derivative of sin(cos(tan))

Answer 13

Chai rule twice

Question 14

Find f’(x) if f=sqrt (x^2 + 1)

Answer 14

Question 15

Pic. Solve before changing original equation

Answer 15

Thus dy/dx = - x/y

Question 16

Solve first change equation setting y one side

Answer 16

Question 17

Y’’ if x^4 + y^4 = 16

Second derivative

Answer 17

Then substitute y’ in and then original equation to get

Y’’= -48 x^2\y^7

Pg 212

Question 18

What does acrsin function mean?

Answer 18

Sin (y) = x

Question 19

d/dx (logb x)

Answer 19

exponentiate to get the b^y

Remember this!

Question 20

d/dx ln x

Answer 20

1/x

b/c ln e = 1

Question 21

d/dx ln g(x)

Answer 21

g’(x) / g(x)

Question 22

d/dx f(x) = ln (|x|)

Answer 22

Question 23

pic

Answer 23

pic

Question 23

Why use “implicit differentiation”?

Answer 23

On graphs that fail the vertical line test, like circles (aka implicit functions where you cannot solve for y explicitly)

x^2 + y^2 = 1

Question 24

pic

Answer 24

Can replace coxy to have function of only x… use scocahtoa in top right

Question 25

differentiate arctan

Answer 25

Question 26

Answer 26

Question 27

Answer 27

Question 28

Answer 28

Notice: variable ^ variable
Then use logarithmic differentiation

Question 29

What is the limit definition of e?

Answer 29

Question 30

d/dx b^g(x)

Answer 30

b^g(x) * ln(b) * g’(x)