Find the derivative of f(x) by using product rule and expanding polynomials

Question

(a) Find the derivative of f(x)=(x2+3)(4x−3) by first expanding the polynomials.

 

Enter the fully simplified expression for f(x) after expanding the polynomials.

 f(x)=

Enter the derivative of f(x).

 f′(x)=

 

(b) Find the derivative of f(x)=(x2+3)(4x−3) by using the product rule. Let g(x)=x2+3 and h(x)=4x−3.

 g′(x)=

h'(x) =

f'(x) =

Answer 1

Expansion of the two polynomial terms results from "FOIL"ing them...

f(x) = (x^2 + 3)(4x - 3)

f(x) = x^2*4x - 3x^2 + 12x - 9 = 4x^3 - 3x^2 + 12x - 9

d/dx f(x) = d/dx ( 4x^3 - 3x^2 + 12x - 9 ) = d/dx (4x^3) - d/dx (3x^2) + d/dx (12x) - d/dx (9)

d/dx f(x) = 4*3*x^2 - 3*2*x + 12 - 0 = 12x^2 - 6x + 12

Using the product rule : define g(x) = x^2 + 3, h(x) = 4x - 3

d/dx f(x) = f'(x) = g(x)*h'(x) + g'(x)*h(x)

g'(x) = 2x

h'(x) = 4

Therefore --> d/dx f(x) = (x^2 + 3)*(4) + (2x)*(4x - 3) = 4x^2 + 12 + 8x^2 - 6x = 12x^2 - 6x + 12