Page 89 - Calculus Demystified

Foundations of Calculus
CHAPTER 2
76
2
2
Math Note: Calculate (d/dx)(x /x) using the quotient rule. Of course x /x = x,
and you may calculate the derivative directly. Observe that the two answers are
the same. The calculation confirms the validity of the quotient rule by way of an
example. Use a similar example to confirm the validity of the product rule.
2.5.1 THE DERIVATIVE OF AN INVERSE
An important formula in the calculus relates the derivative of the inverse of a
function to the derivative of the function itself. The formula is
1
[f −1 . (.)
] (t) =
f (f −1 (t))

We encourage you to apply the Chain Rule to the formula f(f −1 (x)) = x to obtain
a formal derivation of the formula (.).
EXAMPLE 2.19
Calculate the derivative of g(t) = t 1/3 .

SOLUTION
3
2
Set f(s) = s and apply formula (.). Then f (s) = 3s and f −1 (t) = t 1/3 .

With s = f −1 (t) we then have
1 1 1 1 −2/3
[f −1 = = = · t .
] (t) =
f (f −1 (t)) 3s 2 3 ·[t 1/3 2 3

]
Formula (.) may be applied to obtain some interesting new derivatives to add
to our library. We record some of them here:
d 1
I. arcsin x = √
dx 1 − x 2
d 1
II. arccos x =−√
dx 1 − x 2
d 1
III. arctan x =
dx 1 + x 2
√
You Try It: Calculate the derivative of f(x) = x. Calculate the derivative of
√
g(x) = k x for any k ∈{2, 3, 4,... }.

2.6 The Derivative as a Rate of Change

If f(t) represents the position of a moving body, or the amount of a changing quan-

tity, at time t, then the derivative f (t) (equivalently, (d/dt)f (t)) denotes the rate
of change of position (also called velocity) or the rate of change of the quantity.