Differentiation Confusion

Question

If we expand (3x^2 - 1)^4 we get,

 

(3x^2 - 1)^4 = 81x^8 - 108x^6 + 54x^4 - 12x^2 + 1

 

If we derive (3x^2 - 1)^4 we get, (via chain rule)

 

24x(3x^2 - 1)^3

 

However if we derive (81x^8 - 108x^6 + 54x^4 - 12x^2 + 1) we get,

 

27x^7 - 27x^5 + 9x^3 - x

 

Why are the derivatives different for two functions that are equal? Why does the result change when you expand as opposed to using the chain rule?

 

 

 

Answer 1

The derivative of

 

81x⁸ - 108x⁶ + 54x⁴ - 12x² + 1

 

is

 

648x⁷ - 648x⁵ + 216x³ - 24x

 

Now expand 

 

24x (3x² - 1)³ 

 

and get

 

24x [27x⁶ - 3(9x⁴) + 3(3x²) - 1] = 648x⁷ - 648x⁵ + 216x³ - 24x

 

They are the same.  It looks like you have an error in the differentiation of the expanded binomial.  Please take another look at it, per Kenneth's suggestion.

Answer 2

Check your work carefully; whether you differentiate using chain rule (without expanding the binomial) or you expand first and then differentiate the polynomial, the results must be the same...but to show that you might have to expand the version of the derivative that you got in the first instance in order to put it also into the form of a polynomial.