Product Rule

In this lesson we will learn how to differentiate the product of two functions. 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), that is, the product of u(x) and 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

 

Separating terms gives us:

 

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

 

We can add -u(x+Δx)v(x) + v(x)u(x+Δx), which is really 0, to the limit above:

 

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

 

Let's look at the second term. If we factor out v(x) we get:

 

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

 

which is equal to v(x) • du/dx. The same can be done to the first term
to get u(x) • dv/dx.
 
As a general rule:

 

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

 

Find dy/dx for the following:

 

y = (6x3+1) • 4x5

 

Let u(x) = 6x³+1 and v(x) = 4x⁵. Then:

 

dy           d(4x5)       d(6x3+1)
   = (6x3+1)•       + 4x5•         = (6x3+1)•(20x4) + 4x5•(18x2)
dx             dx           dx

   = 180x7 + 20x4 + 72x7 = 192x7 + 20x4