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# 11.2 Some General Differentiation Theorems.

11.12   Theorem (Sum rule for derivatives.) Let be real valued functions with domain and domain , and let . Suppose and are differentiable at . Then , and are differentiable at , and

Proof: We will prove only the first statement. The proofs of the other statements are similar. For all we have

By the sum rule for limits of functions, it follows that

i.e.

11.13   Examples. If

then

so

If , then , so

If , then , so

11.14   Exercise. Calculate the derivatives of the following functions:
a)
b)
c)
d)
e)
f)
g)

11.15   Exercise. A Calculate
a)
b)
. Here and are all constants.
c)

11.16   Theorem (The product rule for derivatives.) Let and be real valued functions with and . Suppose and are both differentiable at . Then is differentiable at and

In particular, if is a constant function, we have

Proof: Let be a generic point of . Then

We know that and . If we also knew that , then by basic properties of limits we could say that

which is what we claimed.

This missing result will be needed in some other theorems, so I've isolated it in the following lemma.

11.17   Lemma (Differentiable functions are continuous.) Let be a real valued function such that . Suppose is differentiable at a point . Then . (We will define  continuous" later. Note that neither the statement nor the proof of this lemma use the word  continuous" in spite of the name of the lemma.)

Proof:

Hence by the product and sum rules for limits,

11.18   Example (Leibniz's proof of the product rule.) Leibniz stated the product rule as

His proof is as follows:

is the difference between two successive 's; let one of these be and the other into ; then we have

the omission of the quantity which is infinitely small in comparison with the rest, for it is supposed that and are infinitely small (because the lines are understood to be continuously increasing or decreasing by very small increments throughout the series of terms), will leave .[34, page 143]
Notice that for Leibniz, the important thing is not the derivative, , but the infinitely small differential, .

11.19   Theorem (Derivative of a reciprocal.) Let be a real valued function such that . Suppose is differentiable at some point , and . Then is differentiable at , and

Proof: For all

It follows from the standard limit rules that

11.20   Theorem (Quotient rule for derivatives.) Let be real valued
functions with and . Suppose and are both differentiable at , and that . Then is differentiable at , and

11.21   Exercise. A Prove the quotient rule.

11.22   Examples. Let

Then by the quotient rule

Let . Then by the product rule

(since ).

The calculation is not valid at (since is not differentiable at , and we divided by in the calculation. However is differentiable at  since , i.e., . Hence the formula

is valid for all .

Let . Consider to be a product where and . Then we can apply the product rule twice to get

11.23   Exercise (Derivatives of tangent, cotangent, secant, cosecant.) We define functions tan, cot, sec, and csc by

The domains of these functions are determined by the definition of the domain of a quotient, e.g. . Prove that

(You should memorize these formulas. Although they are easy to derive, later we will want to use them backwards; i.e., we will want to find a function whose derivative is . It is not easy to derive the formulas backwards.)

11.24   Exercise. A Calculate the derivatives of the following functions. Simplify your answers if you can.
a)
.
b)
(here are constants).
c)
.
d)
.

11.25   Exercise. Let , , , and be differentiable functions defined on .

a) Express in terms of , , , , and .

b) On the basis of your answer for part a), try to guess a formula for . Then calculate , and see whether your guess was right.

Next: 11.3 Composition of Functions Up: 11. Calculation of Derivatives Previous: 11.1 Derivatives of Some   Index
Ray Mayer 2007-09-07
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