Question

How do you differentiate `g(x) = (2x^2 + 4x - 3) ( 5x^3 + 2x + 2)` using the product rule?

Answer 1

`g' (x)=d/dxg(x)=50x^4+80x^3-33x^2+24x+2`

Explanation:

For derivative of product , we have the formula
`d/dx(uv)=u dv/dx + v du/dx`

From the given `g(x)=(2x^2+4x-3)(5x^3+2x+2)`

We let `u=2x^2+4x-3` and `v=5x^3+2x+2`

`d/dx(g(x))=(2x^2+4x-3) d/dx(5x^3+2x+2)+(5x^3+2x+2) d/dx(2x^2+4x-3)`

`d/dx(g(x))=(2x^2+4x-3) (15x^2+2)+(5x^3+2x+2) (4x+4)`

Expand to simplify

`d/dx(g(x))=(2x^2+4x-3) (15x^2+2)+(5x^3+2x+2) (4x+4)`

`d/dx(g(x))=30x^4+4x^2+60x^3+8x-45x^2-6+20x^4+20x^3+8x^2+8x+8x+8`

Combine like terms

`d/dx(g(x))=50x^4+80x^3-33x^2+24x+2`

God bless...I hope the explanation is useful.