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