Derivative help

Question

f(x)=2(x⁷+3x⁴+4x²+1)(x⁵-3x³-7x)

 

What is the simplest way of doing this ? Im confused by the 2. 

Answer 1

 

If you use the product rule, you can let one term be 2(x⁷+3x⁴+4x²+1)

 

f'(x) = "first * derivative of 2nd  + 2nd * derivative of 1st"

 

f'(x) = 2(x⁷+3x⁴+4x²+1)(5x⁴-9x²-7) + (x⁵-3x³-7)(2)(7x⁶+12x³+8x)    etc..    

 

 

You can think of the 2(x... ) multiplication as requiring another product rule in itself.   After doing enough of these you will be able to quickly see that the 2 can just go along for the ride, so-to-speak:

 

Example:   g(x) = 2(x⁷+3x⁴+4x²+1)

                  g'(x) = 2(7x⁶+12x³+8x) + (x⁷+3x⁴+4x²+1)•(0)

 

The last value, the bold 0, is the derivative of '2' with respect to x, so the entire 2nd term vanishes and you are left with the same result (also in bold) we arrived at above.

 

 

Also note that the rule   d{k•h(x)} = k•d{h(x)}  could have been applied directly when finding the derivative of the 2(x⁷ …)  term.

 

Answer 2

f'(x)

=

24 x^ 11 - 60 x^9 + 54 x^8 - 112 x^7 - 70 x^6-320x^4 - 18x^2 - 14