Question

question find f(x) of the following equation. f(x)=(3x^2 - x + 5)(x^2+3x - 9) select the correct answer below: o f(x)=12x^3 + 24x^2 - 50x + 24 o f(x)=12x^2 + 24x - 50 o f(x)=3x^4 + 6x^3 - 25x^2 + 24x - 40 o f(x)=6x - 1

Explanation:

Step1: Apply product - rule

If $f(x)=u(x)v(x)$, then $f^{\prime}(x)=u^{\prime}(x)v(x)+u(x)v^{\prime}(x)$. Let $u(x)=3x^{2}-x + 5$ and $v(x)=x^{2}+3x - 9$.

Step2: Find $u^{\prime}(x)$

Differentiate $u(x)=3x^{2}-x + 5$ using the power - rule $\frac{d}{dx}(ax^{n})=nax^{n - 1}$. So $u^{\prime}(x)=6x-1$.

Step3: Find $v^{\prime}(x)$

Differentiate $v(x)=x^{2}+3x - 9$ using the power - rule. So $v^{\prime}(x)=2x + 3$.

Step4: Calculate $f^{\prime}(x)$

$f^{\prime}(x)=(6x - 1)(x^{2}+3x - 9)+(3x^{2}-x + 5)(2x + 3)$
$=(6x\cdot x^{2}+6x\cdot3x-6x\cdot9 - 1\cdot x^{2}-1\cdot3x + 1\cdot9)+(3x^{2}\cdot2x+3x^{2}\cdot3-x\cdot2x-x\cdot3 + 5\cdot2x+5\cdot3)$
$=(6x^{3}+18x^{2}-54x - x^{2}-3x + 9)+(6x^{3}+9x^{2}-2x^{2}-3x + 10x + 15)$
$=(6x^{3}+17x^{2}-57x + 9)+(6x^{3}+7x^{2}+7x + 15)$
$=6x^{3}+17x^{2}-57x + 9+6x^{3}+7x^{2}+7x + 15$
$=12x^{3}+24x^{2}-50x + 24$

Answer:

$f^{\prime}(x)=12x^{3}+24x^{2}-50x + 24$