Question

find f(x).
f(x)=(9x + 7)(3x - 4)
f(x)=
(type an exact answer.)

Explanation:

Step1: Apply product - rule

The product - rule states that if $y = u(x)v(x)$, then $y^\prime=u^\prime(x)v(x)+u(x)v^\prime(x)$. Let $u(x)=9x + 7$ and $v(x)=3x - 4$.

Step2: Find $u^\prime(x)$ and $v^\prime(x)$

Differentiate $u(x)=9x + 7$ with respect to $x$. Using the power - rule $\frac{d}{dx}(ax + b)=a$, we get $u^\prime(x)=9$. Differentiate $v(x)=3x - 4$ with respect to $x$, we get $v^\prime(x)=3$.

Step3: Substitute into product - rule formula

$f^\prime(x)=u^\prime(x)v(x)+u(x)v^\prime(x)=9(3x - 4)+(9x + 7)\times3$.

Step4: Expand and simplify

First, expand: $9(3x - 4)=27x-36$ and $(9x + 7)\times3 = 27x+21$. Then $f^\prime(x)=27x-36 + 27x+21$. Combine like terms: $f^\prime(x)=(27x+27x)+(-36 + 21)=54x-15$.

Answer:

$54x - 15$