Derivative of Monomials

In this lesson we will learn how to find the derivative of a term such as axⁿ. Let's derive the rule now.
 
By definition:

 

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

 

Now, let's let f(x) = axⁿ. Then, f(x+Δx) = a(x+Δx)ⁿ.

 

dy        a(x+Δx)n - axn
   = lim                
dx   Δx→0       Δx

 

Using the binomial theorem, we can expand a(x+Δx)ⁿ into axⁿ + anx^n-1Δx ... + aΔxⁿ.

 

dy        axn + anxn-1Δx + ... + aΔxn - axn
   = lim                                  
dx   Δx→0               Δx

 

Notice the axⁿ and -axⁿ terms cancel. We can factor out the Δx terms from the numerator and cancel it with the Δx term in the denominator.

 

dy
   = lim  anxn-1 + ... + aΔxn-1
dx   Δx→0

 

As Δx approaches zero, all the terms containing Δx will approach zero. The limit of the above expression is anx^n-1.
 
The general rule for the derivative of a monomial can be written as:

 

d(axn)
       = anxn-1
  dx

 

Find:

 

d(9x6)
      
  dx

 

Using the general rule for the derivative of a monomial:

 

dy   d(9x6)
   =        = 9 * 6 * x6-1 = 54x5
dx     dx