**Differentiation of the Sum, Difference, Product and Quotient of Two Functions**

**The**orem:

If are differentiable at , where I is an open interval in , then so are , fg at . Furthermore, is differentiable at if . We have

(a) derivative of at is

(b) derivative of at is

(c) derivative of at is if

The proofs are straightforward and therefore omitted.

We also have

**Theorem (Chain Rule):**

Let I and J be two intervals in and , and be differentiable at and respectively. Then, is differentiable at and

Note that is defined as .

**Proof.**

Let us write so that by the continuity of f at , we have that as , . Since g is differentiable at , we have

Here as . Again, since f is differentiable at , we have

.

Here, as . Thus, we have

, which equals

, which in turn, equals

where

.

Surely, as and hence,

.

The above result is also often called the Chain Rule.

**Differential Notation of Leibniz**

For a differentiable function f, if we write and , then we get

The expression

is often written as .

It is **NOT** true that is the quotient of limits of and of because both of them tend to zero. It should rather be thought of as an operator (or operation) is the operation of differentiation operating on the variable y so that we have

.

The operator has the property and

, and for f and g, two differentiable functions with the domain of g containing the range of f, if we write and , we have , then the chain rule gives . In the case, when so that we write, following the German mathematician Leibnitz,

,

and dy and dx are called the differentials of y and x respectively.

**Remark:**

Let be a function. We say that f is differentiable at if we can find numbers A, B depending on only, so that

such that

as

Observe that

and

We call A and B the partial derivatives of f with respect to x and y respectively at , and we write

,

Sometimes, we also write and . Again, as before, and may be thought of as operators which, operating on a function, give its partial derivatives with respect to x and y respectively.

Suppose that is differentiable, and that x and are differentiable functions. Furthermore, let be defined by

It is not difficult to show that (**Exercise!**)

.

In Leibnitz’s notation, it reads .

As an application of this idea, consider the notion of equipotential surface in electrostatics. An electrically charged (infinite) cylindrical conductor is one such and because of the symmetry, it is enough to look at only a horizontal section of the cylinder which is a closed curve, that is, the problem gets reduced to a 2-dimensional one. This equipotential curve is given by the equation , say c, where is the real valued potential function. What is the electric field outside the curve? Using Leibnitz’s notation as described above, we can write .

Since is a constant, the relation above reads as

,

that is, the vector is orthogonal to the tangent vector .

Recalling that the electric field E at a point outside has components and , we conclude that the electric field E is along the outward normal. For example, for a right circular cylinder, the electric field is along the radial direction of every section.

We have seen that if f is differentiable at , then is of smaller order than as .

Writing , we get

which is of smaller order than . In other words, we are claiming that the increment in y is proportional to the increment in x when it is *small*, which is the principle of proportional parts. We have put the word small within asterisks as it is a relative term. Let us consider an example.

Let with . Then, for . So, , , since . This gives us , whereas computation on a hand calculator will give This shows that and thus the differential approximates the increment in correct at least to the second place of decimal.

More later,

Nalin Pithwa