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Finding Derivatives using Formula

The following rules are useful in finding derivatives using formula for the various functions:

 

Rule 1: If f(x) = xn, then f'(x) = nxn-1.

Or if y = xn, then y’  or  {dy}/{dx}  or   {dx^n}/{dx} = nxn-1.

 

Rule 2: If f(x) = kxn, then f'(x) = knxn-1.

Or if y = kxn, then y’  or  {dy}/{dx}  or   {dkx^n}/{dx} = nkxn-1.

In general,  {d}/{dx}kx^n  =  k  {d}/{dx}x^n

So  {d}/{dx}kf(x)  =  k f'(x)

 

Rule 3: When an equation contains the sum of two functions, then the derivative of this equation is the sum of the derivatives of the two functions.

 {d}/{dx}(f(x)~+~g(x))  =   {d}/{dx}f(x)  +   {d}/{dx}g(x)

=  f'(x) + g'(x)

Proof for this rule:

 {d}/{dx}(f(x)~+~g(x))  =  lim{h right 0}{{f(x+h)~+~g(x+h)~-~f(x)~-~g(x)}/{h}}

 {d}/{dx}(f(x)~+~g(x))  =  lim{h right 0}({{{f(x+h)~-~f(x)}/{h}~+~{g(x+h)~-~g(x)}/{h}}})

=  lim{h right 0}{{{f(x+h)~-~f(x)}/{h}~+~lim{h right 0}{{g(x+h)~-~g(x)}/{h}}}}

=  f'(x)  +  g'(x)

 

Rule 4: Similarly the derivation of the difference of two equations is the difference of the derivatives of the two functions.

 {d}/{dx}(f(x)~-~g(x))  =   {d}/{dx}f(x)  –   {d}/{dx}g(x)

 {d}/{dx}(f(x)~-~g(x))  =  f'(x)  –  g'(x)

Proof for this rule:

 {d}/{dx}(f(x)~-~g(x))  =  lim{h right 0}{{f(x+h)~-~g(x+h)~-~f(x)~+~g(x)}/{h}}

 {d}/{dx}(f(x)~-~g(x))  =  lim{h right 0}({{{f(x+h)~-~f(x)}/{h}~-~{g(x+h)~-~g(x)}/{h}}})

=  lim{h right 0}{{{f(x+h)~-~f(x)}/{h}~-~lim{h right 0}{{g(x+h)~-~g(x)}/{h}}}}

=  f'(x)  –  g'(x)

 

Rule 5: The derivative of the product of two functions – known as Product Rule – is given by

 {d}/{dx}(f(x)~*~g(x))  =   ({d}/{dx}f(x))g(x)  +  (f(x)}~({d}/{dx}g(x))

=  g(x) f'(x)  +  f(x) g'(x)

 

 

Rule 6: The derivative of the quotient of two functions – known as Quotient Rule – is given by

 {d}/{dx}({f(x)}/{g(x)})  =   {({d}/{dx}f(x))g(x)~-~(f(x)}~({d}/{dx}g(x))}/{(g(x))^2}

This rule is a modification of the Product Rule by writing

 {d}/{dx}({f(x)}/{g(x)}) as  {d}/{dx}(f(x)~g(x)^-1)

 {d}/{dx}(f(x)~g(x)^-1)  =   {d}/{dx}f(x)~g(x)^-1  +  f(x)  {d}/{dx}g(x)^-1

=  f'(x).(g(x))-1  −  f(x).(g(x))-2.g'(x)

=  {{f}prime{(x)}}/{g(x)}  –  {{f(x)}{g}prime{(x)}}/{(g(x))^2}

{{{f}prime{(x)}}{g(x)}~-~{{f(x)}{g}prime{(x)}}}/{(g(x))^2}

 

In simplistic terms, we can look at the Product Rule and Quotient Rule as follows:

Let u = f(x) and v = g(x).

The product rule can be expressed as (uv) = u’v + uv’, and

The quotient rule can be expressed as {(u/v)}prime  =  {{u}prime{v}~-~{v}prime{u}}/{v^2}

These formulae are referred as Leibnitz rule for differentiation of product and quotient of functions respectively.

 

Now let us look at some examples of differentiation using these formulae.