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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:

f(x) = c, where c is a constant, xn, sin x, cos x, ex and ln x.

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

f'(x)= the limit as h tends to 0 of (f(x+h)-f(x))/h = the limit as h tends to 0 of (c-c)/h = the limit as h tends to 0/h = 0.


Hence

The derivative of a constant is c. , 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

the derivative of x to the power n is n times x to the power n-1

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

(x+h)^n = x^n + n*x^(n-1)*h+Qh^2 ,

where Q is a polynomial in x and h.

Thus

the limit as h tends to 0 of ((x+h)^n-x^n)/h = n*x^(n-1).

Hence for any non-negative integer n

the derivative of x^n with respect to x is n*x^(n-1).

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

For example: (d/dx)(x^0.5)=0.5*x^(0.5-1)=0.5*x(^-0.5). (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.) The limit as h tends to 0 of sinh/h=1           (Rough proof)

(3.) the limit as h tends to 0 of (cosh - 1)/h=0.           (Proof)


The Newton's Quotient for the sin x is then

sinx*(cosh-1)/h+cosx*(sinh/h).

From results (2) and (3) we see that this quotient tends to cos x as h tends to 0. .

Hence

d/dx(sinx)=cosx.

 

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.) The limit as h tends to 0 of sinh/h=1           (Rough proof)

(3.) the limit as h tends to 0 of (cosh - 1)/h=0.           (Proof)

The Newton's Quotient for the cos x is then

cosx*(cosh-1)/h-sinx*s9nh/h.

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

Hence

d/dx(cosx)=-sinx.

 

The derivative of ex

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

the limit as h tends to 0 of (1+h)^(1/h)=e

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

This equality implies that when h is small in magnitude,

(1+h)^(1/h) is approximately equal to e.

That is

1+h is approximately equal to e^h.

Therefore,

d/dx(d^x)=e^x).

Hence

d/dx(d^x)=e^x).

 

 

The derivative of ln x

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

the limit as h tends to 0 of (1+h)^(1/h)=e

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

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

ln e = 1.

Let f(x) = ln x. Then

the newton quotient is 1/h*ln(1+h/x).

Let h/x=h . Thus h = kx and the above becomes

1/x*ln(1+k)^(1/k).

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

the derivative is 1/x.

Therefore

d/dx(lnx)=1/x.

 

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