Download Calculus 050a/b Summary Sheet Derivatives of the trig functions 1

Calculus 050a/b Summary Sheet
Derivatives of the trig functions
1. Facts to remember
Group function
derivative
df
dx
cos x
sec2 x
sec x tan x
f (x)
A
B
C
sin x
tan x
sec x
function
f (x)
cos x
cot x
csc x
derivative
df
dx
− sin x
− csc2 x
− csc x cot x
2. How to remember these: notice the pattern
• The six trig functions break up into three pairs, each one being a function and its partner,
or ‘cofunction’: sine and cosine; tangent and cotangent; secant and cosecant. This is
symmetric, in the sense that the ‘co-cofunction’ is the function itself: for example, the
partner of the sine, the“co-cosine”, is the sine.
• If you remember the derivative of any one of these functions, it is easy to recapture the
derivative for its co-function. Just replace every function in the formula by it’s co-function,
and change the sign. For example:
d
(sec x) = + (sec x) (tan x)
dx
d
reminds us that
(csc x) = − (csc x) (cot x)
dx
remembering
3. Why are these formulas true? How could we prove them?
Memorizing the rules above is not the same thing as knowing why the rules are valid, so let’s
talk a bit about where these formulas come from. To begin, we’ll establish the Group A formulas
d
d
that
sin x = cos x and
cos x = − sin x. Remember the definition(s):
dx
dx
f (x0 ) − f (x)
and also
x →x
x0 − x
f (x + h) − f (x)
= lim
h→0
h
df
= f 0 (x) =
dx
lim
0
The Group A formulas
• The first thing to do is to prove special cases: what are the derivatives of sine and cosine
at x = 0 ? You will find more details at box 2, page 212 and box 3, page 213, of the
Stewart text.
µ
¶
µ
¶
d
sin(0 + h) − sin 0
d
cos(0 + h) − cos 0
sin x
= limh→0
cos x
= limh→0
dx
h
dx
h
x=0
x=0
sin h − 0
cos h − 1
= limh→0
= limh→0
=0
h
h
sin h
= limh→0
=1
h
We will make good use of the two boxed formulas in jsut a moment.
2
Calculus 050a Summary sheet
• Next, we find the derivative of sin at any place x.
d
sin(x + h) − sin x
sin x = lim
h→0
dx
h
(sin x cos h + cos x sin h) − sin x
= lim
(using formula sin(x + h) = . . . )
h→0
h
µ
¶
cos h − 1
sin h
= lim sin x
+ cos x
(just rearrange)
h→0
h
h
µ
¶
µ
¶
cos h − 1
sin h
= sin x lim
+ cos x lim
(using the Limit Laws)
h→0
h→0
h
h
= (sin x) 0 + (cos x) 1 (the boxed formulas above)
= cos x
• Finally, establish the derivative of the cosine function. The steps are similar.
d
cos(x + h) − cos x
cos x = lim
h→0
dx
h
(cos x cos h − sin x sin h) − cos x
= lim
(because cos(x + h) = . . . )
h→0
h
¶
µ
sin h
cos h − 1
= lim cos x
− sin x
h→0
h
h
¶
¶
µ
µ
cos h − 1
sin h
− sin x lim
= cos x lim
h→0
h→0
h
h
= (cos x) 0 − (sin x) 1 = − sin x
The Group B and Group C formulas. These now are easy to get, using the Group A results,
and the ‘quotient rule’”:
µ
¶ g(x) df − f (x) dg
d f (x)
dx
dx .
=
dx g(x)
(g(x))2
For example,
¡
¢
¡
¢
µ
¶
x
x
cos x d sin
− sin x d cos
d sin x
d
dx
dx
tan x =
=
dx
dx cos x
cos2 x
cos x(cos x) − sin x(− sin x)
cos2 x + sin2 x
=
cos2 x
cos2 x
µ
¶2
1
=
= (sec x)2
cos x
=
Here’s one more example.
¡ ¢
¡
¢
µ
¶
x
sin x ddx1 − 1 d sin
d
1
dx
=
dx sin x
sin2 x
sin x(0) − 1(cos x)
=
(1 is a constant so its derivative is 0)
2
sin
x
µ
¶³
1
cos x ´
= −
= − csc x cot x
sin x
sin x
d
csc x =
dx
The other formulas in Groups B and C are left as exercises.