Math 1210-001
Wednesday Feb10
WEB L112
2.4-2.5: Chain rule and trig function derivatives drill day. We’ll go through the first several exercises
together, before letting you test yourselves and practice together on the later problems. The goal is to
develop your ability to do these computations quickly and accurately. You may also have some WebWork
questions.
D x (f(g(x)) =f’(g(xflg’(x)
(chain rule)
(The rate of change of a composition is the product of the rates of change of each function being
composed, at the appropriate input values.)
(The derivative of the composition is the derivative of the outer fi.rnction, evaluated at the inner function
value, times the derivative of the inner function.)
Exercise 1’) D (3 x2 + 7) 100
f’&
Cf ~
lea
tic.’ t~ I,,.
4
(3x. i7)
-
1012 (3st7)
~(,x
Exercise 2~ D~ (3 cos2t + 7)
%t4~Zc~vclt
I~
t(k1~ 3._ti-7
(tjj::.
f ‘(~ti l~ ‘It)
C
Ck.
Gc1ost
(—;ikf.)
—
Exercise 3’ F’ (t) for F(t) = sin(2 t) (understand your answer geometrically by looking at the graph
below, from Monday’s notes).
—
U&)Dlt
~?fi:l
1•
/
(~V9i~c
3ic
it
4
z~
/it
2
t
~cfttP2
‘2~
t~.on-e5p0~J4
‘1-0 ~
5
I
tm~Lc(thcskt
Exercise4) G’(1)ifG(t)=(?+9)3(?—2)4.
fr-3-(~‘tf~j ++~‘
.(tz~zfr* (t~+?)
£~(i)t
3
z.
iop.i. I + Icao.t~
fr4)t(~ *?)
(-‘3(z)
3
a.. ~ni-.gs~vctW-
z
~ a(t).
k
CLcI4.,)
k1t;.C~f.d1
fc4)t k[(k~L6,)*t16)
3 (ttii)’.zE
Exercise5~ fmdf’(3) iff(x)
=
~
+ 1
x+2
k(aC~)
=
~tJt ~
Cx-l-zjt
‘2.
U
-
c,Lt4~
3
~(t’-z) •2~
Coo— cooc, —74cc
2
(xn)t
C~vk(DL
Cx4.2)t
Exercise 6) What’s the symbolic formula for the derivative of a triple composition?
D~fogoh(x)
t~
r
j’(~CL~L,fl •~~(L.~.(,q)
Exercise 7) D~ sin(cos(t2 + 5))
= ~ (~sW-*s)~
~
(±~÷3~)
2-t
Exercise 8) D I sin(t) (leftovers... :-) )
-
-si’’
~JKt ~
1)
Exercise 9)
I~
cos3 (2 t)
c~L~~c4sc-tt$t
J
—
÷1)
.3~zt (-~zt~ ‘2
Things to know page.
Differentiation rules:
D~(x’1) = ii
I ~ S 1.
D~(f(x) +g(x))=D~(f(x)) +D~(g(x))
D(kf(x)) = k D (f(x)) if k is a constant
D~(f(x)g(x)) f’(x)g(x) +f(x)g’(x)
—
D x~(f(x)
g(x) ~,)
f’(x)g(x)—f(x)g’(x)
D(f(g(x)) t=f’(g(xfl.g’(x)
Dcot(x)
Dsec(x)
n
~‘x’5”~
=
(quotientrule)
(ehainrule)
D(sin(x))=cos(x)
D(eos(x)) ‘-sin(x)
Dtan(x)
(power rule; includes the three special cases above.)
(sumrule)
(constant multiple rule)
(productrule)
sec2(x)
—csc2(x)
= sec(x)tan(x)
=_
W5~)wL~.)
Trigonometry identities:
eos2O + sin2O = 1
cos(c(~+ j3) ~=cos(cc)cos(13)
sin(cL)sin(~)
sin(a~+B)=cos(a)sin(J3) +sin(cc)cos(13)
—