Derivative of a Function

Let be a function. Suppose that the limit

exists. The limit is denoted by and is called the derivative of at . In this case way say is differentiable at .

Notice that to say the limit

exists is the same as to say the limit

exists. In this case the two limits are equal.

If then Hence

Since it follows that

Since

we have

for Therefore

for Since we have

Derivative of

It follows from the identity

that

Therefore

Derivative of

It follows from the identity

that

Therefore

Higher-Order Derivatives

Let be the derivative of the function then the derivative of the function is called the second derivative of the function and is denoted by A derivative of the second derivative is called the third derivative of the function It is denoted by Similarly we can define the fourth order fifth order and so forth. A derivative of the nth order is symbolized by and is sometimes also written

The above formulae show that the function has derivatives of all orders and that

This formula is sometimes written

Exercise: Obtain the general formula for the nth order derivative of the function

Let Then







In general we see that the nth order derivative takes the form

Exercise: Find the general formula for the nth derivative of function

The Coefficients of a Polynomial Function

Let be a polynomial function of the form

Then

From

we have

Taking derivatives, we have

therefore

Taking derivatives again, we have

therefore

Continuing this way we see that the coefficients can be obtained from the formula

In the same fashion, we see that if the polynomial were expressed in the form

then the coefficients can be obtained from the formula

Linear Combinations of Functions of the Form

Exercise: Apply the Expand command of Maple to express these functions as linear combinations of functions of the form

(a)

(b)

(c)

(d)

Exercise: Find the nth derivative of each of the above functions.

Exercise: Find the nth order derivative of the function

Leibniz Rule

These are the nth derivative of the product of two functions

In general, we have

Rules for taking the Derivative

Suppose that and exist Then

(i)

(ii)

(iii) ( for scalars

If in addition that for all then

(iv)

(v)

(vi) Let be a polynomial function. Then

Exercise: prove each of the above statements.

Chain Rule

If is differentiable at and is differentiable at then the composition is differentiable at and

Proof. Define a function as follows.

We claim that is continuous at

Let be given. Since is differentiable at there exists such that

Since is differentiable at it follows that is continuous at so there exists such that

Now fix any with We show that

Case 1): Then

Write Then and and so

therefore

Case 2): Then so clearly

We have proved that

Now for we have

even if (both sides are zero!) Therefore