Proofs of Differentiation

Theorem:

If a constant, then .

Proof:

Theorem:

If , where n is a rational number, then

Proof:

We will prove this theorem for all positive integers. The proof for any rational number will be discussed later.

Theorem:

If have derivatives and , then

Proof:

Theorem:

If and are any two functions which have a derivative, and if then,

Proof:

Corollary:

If where c is a constant, then,

.

Lemma:

Let f be the function defined by . Then, .

Proof:

Theorem:

If are any two functions which have derivative and , then,

Proof:

The Chain Rule:

Suppose we have two functions f and g, such that g is a function of x and f is a function of g(x). The derivative of the composite function with respect to x is obtained by the relation:

. Or,

.

Proof:

We know that .

We’ll put . Then, we can proceed as following:

Corollary:

If , n is an integer, then,

Theorem:

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Proof:

To prove that :

Let .

To prove that :

Since , we can put, then

To prove that :