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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 ∆_h f = f(a +h) − f(a), etc., so that

f′(a) = lim h → 0  ∆h f h ,

etc.,

(f ·g)(a + h) = (f(a) + ∆h f) ·(g(a) + ∆h g) = f(a)·g(a) + ∆h f ·g(a) + f(a) · ∆h g + ∆h f ·∆h g, ∆h (f·g) = (f ·g)(a + h) − f(a)·g(a) = ∆h f ·g(a) + f(a) · ∆h g + ∆h f ·∆h g, lim h → 0  ∆h (f·g) h = lim h → 0  ∆h f h ·g(a) + f(a) · lim h → 0  ∆h g h + lim h → 0  ∆h f h · lim h → 0  ∆h 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 (♠)

for all a.

Derivative of a Quotient

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

⎛ ⎝ f g ⎞ ⎠ ′   (a) = g(a)·f′(a) − f(a) ·g′(a) |g(a)|2 .

An alternate form:

⎛ ⎝ u v ⎞ ⎠ ′   = v u′ − v′ u v2 .

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

⎛ ⎝ u v ⎞ ⎠ ′   :                                                            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:

⎛ ⎝ u v ⎞ ⎠ ′   = ⎧ ⎨ ⎩ u · ⎛ ⎝ 1 v ⎞ ⎠ ⎫ ⎬ ⎭ ′   = u′ v − u · ⎛ ⎝ v′ v2 ⎞ ⎠ ,

(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(♥(…)) dx : f′(                                                        ) (deriv of outside function f) → f′(♥(…)) (evaluate at inside function) → f′(♥(…))· (TIMES) → f′(♥(…))· ⎛ ⎝ d(♥(…)) dx ⎞ ⎠ (deriv of inside) (do it again if necessary)

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