3-6

1. (Warm-up)A 12-foot ladder is leaning against a wall. Let x denote the distance of the base of the ladder

from the wall, and let ✓ be the angle between the ladder and the wall.

(a) How fast does the angle ✓ change with respect to x?

(b) I compute that d✓/dx ⇡ 0.1 when x = 7. What does this mean in language your parents can under-

stand? Feel free to express your answer in terms of degrees instead of radians.

2. Vera says she is not a huge fan of logarithms so rewrites the function y = lnx as x = ey . Is this ok?

3. Finddy

dximplicitly for x = ey and write your answer in terms of x.

4. Finddy

dximplicitly for x = ay and write your answer in terms of x.

Congratulations, you just derived the formulas for the derivatives of logarithms.

UAF Calculus 1 1 3-6

algebra 1-asia ,.%[ Fifa

Doesthis/ make

sense?

÷# III?¥÷f¥I÷⇐IIL÷i*÷tra%

.

Oilradnfljoonaia = 1¥ in"g°=6° .

So, parents, when

the base of

the ladder is 7ft from the

wall and being pushed away from the wall,the angle with

the wall is increasing at a rate of about 6°

per foot .

yes .

This is a demonstration of the definitionof lnx as inverse of ex

.

( ie : How do you find

the inverse of y=fCa ? Switch xty and solve for y . )

I = et . dotx.

so o¥×= ety = ¥ txt

I = dna) . also day so E¥=@g÷aJ=@a÷afa×=kn¥×

L (

u

Yay !

Using the formulas you just derived (and possibly the chain rule and/or the quotient rule and/or the

product rule...) find the derivatives of each of the following:

5. f(x) = (lnx)7/2

6. f(x) = ln(px)

7. f(x) = ln(3x+ 1)

8. Consider y =⇣

x2�23�x

⌘3

(a) Without actually taking the derivative, list the rules you would need to do so.

(b) Use rules of logarithms, expand the right-hand side and then take the derivative.

UAF Calculus 1 2 3-6

f 'lx7=¥(|nD% . } =

74×51 4ham rule)

ZX

= In ( x "2)=tzlnx ;so ftxktzxtztx

£rulesabout logs.

( chain rule )

f 'lx7= 3¥, ° 3=3×3+7

ATNEEDY° Chain rule

. quotient rate

y=ln(y5÷)3) =3 In ( 5¥) . 3[ incest - ING - xD

o¥=3f¥ - ÷d=×¥ts÷

ttoocanbexpoted:

Y = Cos*)× -why

don't anyprevious

ruleswork ?

Newpwbtem

In y= In [@sD×]

lny = × In ( cosx )

( take derivative implicitly

ty . y'

= 1 ° In (

cosDtXwsz.slnXEYxt-yflnCcosD-xtanxTt@sxYGnCcosxhxtanxTSketchyFxl.ca

nyou give

a formula for junt the LHS ?

f ( × ) =In (a)

f'

( D= ±× . text