Page 85 - Calculus Demystified

Foundations of Calculus
CHAPTER 2
72
2
3
and the derivative of x is 3x . The rule just enunciated is a generalization of these
facts, and is established in just the same way.
B Derivatives of Trigonometric Functions: The rules for differentiating sine
and cosine are simple and elegant:
d
1. sin x = cos x.
dx
d
2. cos x =− sin x.
dx
We can find the derivatives of the other trigonometric functions by using these two
facts together with the quotient rule from above:
d d sin x cos x(d/dx) sin x − sin x(d/dx) cos x
3. tan x = =
dx dx cos x (cos x) 2
2
(cos x) + (sin x) 2 1
2
= = = (sec x) .
(cos x) 2 (cos x) 2
Similarly we have
d
2
4. cot x =−(csc x) .
dx
d
5. sec x = sec x tan x.
dx
d
6. csc x =− csc x cot x.
dx
C Derivatives of ln x and e : We conclude our library of derivatives of basic
x
functions with
d x x
e = e
dx
and
d 1
ln x = .
dx x
We may apply the Chain Rule to obtain the following particularly useful general-
ization of this logarithmic derivative:

d ϕ (x)
ln ϕ(x) = .
dx ϕ(x)
Now it is time to learn to differentiate the functions that we will commonly
encounter in our work. We do so by applying the rules for sums, products, quotients,
and compositions to the formulas for the derivatives of the elementary functions.