Derivative of Sum of Monomials

In this lesson we will learn how to differentiate a sum of monomials,
such as u(x) + v(x). First, let's derive the rule.
 
We'll begin with the definition of a derivative:

 

dy        f(x+Δx) - f(x)
   = lim                
dx   Δx→0       Δx

 

Now, let f(x) = u(x) + v(x). Then, f(x+Δx) = u(x+Δx) + v(x+Δx).

 

dy        u(x+Δx)+v(x+Δx) - [u(x)+v(x)]
   = lim                               
dx   Δx→0             Δx

 

Now let's distribute the minus sign and switch the positions of the second and third terms.

 

dy        u(x+Δx)-u(x) + v(x+Δx)-v(x)
   = lim                             
dx   Δx→0             Δx

          u(x+Δx)-u(x)        v(x+Δx)-v(x)
   = lim               + lim              
     Δx→0       Δx       Δx→0       Δx

 

We can see that this is actually the sum of du(x)/dx and dv(x)/dx.
 
As a general rule:

 

d(u(x)+v(x))   d(u(x))   d((v)x))
             =         +         
     dx          dx         dx

 

Find:

 

d(6x2 + 9x6)
            
     dx

 

We will use the general rule for the derivative of the sum of monomials.

 

d(6x2+9x6)    d(6x2) + d(9x6)
           =                
    dx         dx       dx

           = 6•2•x2-1 + 9•6•x6-1 = 12x + 54x5