How do you differentiate y=(x^4+3)(-4x^5+5x^4+5) using the product rule?

1 Answer
Apr 3, 2017

y'=-4x^3(9x^5-10x^4+15x-20)

Explanation:

Denote f(x)=x^4+3 and g(x)=-4x^5+5x^4+5 so that y is of the form f.g.

The derivative of y with respect to x according the product rule is then f'.g+f.g':

y' =f'.g+f.g'
y'=(4x^3)(-4x^5+5x^4+5)+(x^4+3)(-20x^4+20x^3)
y'=-16x^8+20x^7+20x^3-20x^8+20x^7-60x^4+60x^3
y'=-36x^8+40x^7-60x^4+80x^3
y'=-4x^3(9x^5-10x^4+15x-20)