How do you find the simplified form of the difference quotient for the function:

f(x) = ax^4

Please help me with this problem if you can.

The difference quotient is just   (  f(x+h)-f(x)   )/h

So just the numerator of the difference quotient f(x+h)-f(x) will yield (if you remember how to expand (x+h) to the 4th power, it is not x^4 + h^4 )

f(x+h)-f(x)  =  a(x+h)^4 -  ax^4
         =  a (x^4 + 4 x^3 h + 6 x^2 h^2 + 4 x h^3 + h^4 ) - ax^4
         =  a x^4 + 4a x^3 h + 6a x^2 h^2 + 4a x h^3 + ah^4  - ax^4
         =  4a x^3 h + 6a x^2 h^2 + 4a x h^3 + ah^4
         =  (4a x^3  + 6a x^2 h + 4a x h^2 + ah^3 ) h    (since every term now has an h)

Now I shall leave it to you to divide that by h (do some cancellation and get a final simplified form of the
difference quotient of f(x) = ax^4.


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