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Tangents, Derivatives and Differentiation

Derivation of The Basic Derivative Rules

Most of the functions that we have to deal with when finding derivatives are combinations of the following elementary functions:

We now try to find the derivative of each of these elementary functions.

The derivative of f(x) = c, where c is a constant

Let f(x) = c, where c is a constant. Then

Hence

, for any constant c

.

The derivative of xⁿ

From Example 2, the examples found in the "More" page following Example 2, and Exercise 1 you may have guessed that

for any integer or fractional number n. In fact this is true for any real number n.

Let us prove this.

Proof: Let n be any non-negative integer. From the binomial theorem, we have

,

where Q is a polynomial in x and h.

Thus

Hence for any non-negative integer n

This is known as the power rule and is in fact true for any real number n.

For example: (As proved in Exercise 1.)

The derivative of sin x

To find the derivative of f(x) = sin x, we first need a couple of results:

(1.) sin(x+h) = sin x cos h +cos x sin h           

(2.)           (Rough proof)

(3.)           (Proof)

The Newton's Quotient for the sin x is then

From results (2) and (3) we see that this quotient tends to cos x as .

Hence

The derivative of cos x

To find the derivative of f(x) = cosx, we first need a couple of results:

(1.) cos(x+h) = cosx cos h − sinx sin h          

(2.)           (Rough proof)

(3.)           (Proof)

The Newton's Quotient for the cos x is then

From results (2) and (3) we see that this quotient tends to −sin x as .

Hence

The derivative of e^x

To find the derivative of f(x) = e^x, we recall the definition of Euler’s number e:

where e is a non-fractional number roughly equal to 2.718281828459045235.

This equality implies that when h is small in magnitude,

That is

Therefore,

Hence

The derivative of ln x

To find the derivative of f(x) = ln x, we again recall the definition of Euler’s number e:

where e is a non-fractional number roughly equal to 2.718281828459045235.

Note that ln x is the inverse of the exponential function e^x. Thus ln e^x = x for all x. In particular,

ln e = 1.

Let f(x) = ln x. Then

Let . Thus h = kx and the above becomes

As h→0, k→0. Hence by the definition of e, from the above we have

Therefore

<< The Derivative  |  Differentiation Index |  The Rise and Fall of a Graph >>