MthT 430 Notes Chapter 10 Differentiation

The purpose of learning differentiation technique is to develop formulas for almost any function defined by a formula.

From the Chapter 5, Limit Theorems, we easily obtain for sums (differences) of differentiable functions:

Theorem 3. If f and g are differentiable at a, then f ±g is differentiable at a, and

(f ±g)¢(a) = f¢(a) ±g¢(a).

More briefly,

Theorem 3. If f and g are differentiable [on a set I], then f ±g is differentiable [on I], and

(f ±g)¢ = f¢±g¢.

Product Rule for Differentiation

Theorem 4. If f and g are differentiable at a, then

(f ·g)¢(a) = f¢(a) ·g(a) + f(a) ·g¢(a).

Proof.p. 167.

The idea behind the proof: Let D_h f = f(a +h) - f(a), etc., so that f¢(a) = lim_{h ® 0}[(D_hf)/h], etc.,

(f ·g)¢(a + h) = (f(a) + Dh f) ·(g(a) + Dh g) = f(a)·g(a) + Dh f ·g(a) + f(a) · Dh g + Dh f ·Dh g, Dh (f·g) = (f ·g)(a + h) - f(a)·g(a) = Dh f ·g(a) + f(a) · Dh g + Dh f ·Dh g, lim h ® 0  Dh (f·g) h = lim h ® 0  Dh f h ·g(a) + f(a) · lim h ® 0  Dh g h + lim h ® 0  Dh f h · lim h ® 0  Dh g = f¢(a) ·g(a) + f(a) ·g¢(a).

The derivative of the n^th power function is given by:

Theorem 6. If f(x) = xⁿ for some natural number n, then

f¢(a) = n an-1 (ª)

for all a.

Derivative of a Quotient

Theorem 7. If g is differentiable at a, and g(a) ¹ 0, then 1/g is differentiable at a, and

(1/g)¢(a) = - g¢(a) (g(a))2 .

Proof. p. 167. An alternate proof is to observe that 1/g is the composition of the oneover function with g and use the chain rule .

Theorem 8 (Quotient Rule). If f and g is differentiable at a, and g(a) ¹ 0, then f/g is differentiable at a, and

æ è f g ö ø ¢   (a) = g(a)·f¢(a) - f(a) ·g¢(a) |g(a)|2 .

An alternate form:

æ è u v ö ø ¢   = v u¢ - v¢ u v2 .

From an old calculus book (Robert Bonic, et al., Freshman Calculus):

æ è u v ö ø ¢   :                                                            v2 (write vinculum over v2) ® v                                                         v2 (write v again before you forget) ® v u¢ - v¢ u v2 (fill in the rest)

Some prefer the forms:

æ è u v ö ø ¢   = ì í î u · æ è 1 v ö ø ü ý þ ¢   = u¢ v - u · æ è v¢ v2 ö ø ,

(u v-1)¢ = u¢v-1 + (-1) u v¢ v-2 . (§)

The form (§) is particularly convenient for repeated derivatives.

Chain Rule

Theorem 9 (Chain Rule). If g is differentiable at a, and f is differentiable at g(a), then f°g is differentiable at a and

(f °g)¢(a) = f¢(g(a)) ·g¢(a).

Other forms of the Chain Rule

Leibniz notation:

y = y(u), dy dx = dy du   du dx .

Leibniz (mixed) notation:

df(u(x)) dx = f¢(u(x))· du dx .

Function notation:

(f °g)¢ = (f¢°g)·g¢

One layer at a time:

df(©(¼)) dx : f¢(                                                        ) (deriv of outside function f) ® f¢(©(¼)) (evaluate at inside function) ® f¢(©(¼))· (TIMES) ® f¢(©(¼))· æ è d(©(¼)) dx ö ø (deriv of inside) (do it again if necessary)

---

File translated from T_EX by T_TH, version 3.79.
On 29 Nov 2007, 09:38.