Simplify the expression representing the derivative of a function.
-$$f ^ { \prime } ( x ) = \left( 4 x ^ { 4 } - 3 x ^ { 3 } + 6 x - 5 \right) ( 6 x ) + ( x - 3 ) \left( 6 x ^ { 2 } - 6 \right)$$
A) $24 x ^ { 5 } - 18 x ^ { 4 } + 6 x ^ { 3 } - 36 x ^ { 2 } - 36 x + 18$
B) $24 x ^ { 5 } - 18 x ^ { 4 } + 6 x ^ { 3 } + 18 x ^ { 2 } - 36 x + 18$
C) $24 x ^ { 5 } - 18 x ^ { 4 } - 6 x ^ { 3 } + 18 x ^ { 2 } - 36 x - 18$
D) $24 x ^ { 5 } - 18 x ^ { 4 } - 18 x ^ { 2 } - 30 x + 18$

Question

Quizplus · Accepted Answer

To simplify the given expression, apply the distributive property (also known as FOIL for binomials) to each product:1. For the first product $(4x^4 - 3x^3 + 6x - 5)(6x)$, multiply each term inside the parentheses by $6x$:- $4x^4 \cdot 6x = 24x^5$- $-3x^3 \cdot 6x = -18x^4$- $6x \cdot 6x = 36x^2$- $-5 \cdot 6x = -30x$2. For the second product $(x - 3)(6x^2 - 6)$, distribute each term inside the first parentheses across the second:- $x \cdot 6x^2 = 6x^3$- $x \cdot -6 = -6x$- $-3 \cdot 6x^2 = -18x^2$- $-3 \cdot -6 = 18$3. Combine like terms from both products:- $24x^5$ (no other $x^5$ terms)- $-18x^4$ (no other $x^4$ terms)- $36x^2 + 6x^3 - 18x^2 = 6x^3 + 18x^2$- $-30x - 6x = -36x$- $18$ (constant term)So, the simplified expression is $24x^5 - 18x^4 + 6x^3 + 18x^2 - 36x + 18$, which matches choice B.

Simplify the Expression Representing the Derivative of a Function f′(x)=(4x4−3x3+6x−5)(6x)+(x−3)(6x2−6)f ^ { \prime } ( x ) = \left( 4 x ^ { 4 } - 3 x ^ { 3 } + 6 x - 5 \right) ( 6 x ) + ( x - 3 ) \left( 6 x ^ { 2 } - 6 \right)f′(x)=(4x4−3x3+6x−5)(6x)+(x−3)(6x2−6)

Simplify the Expression Representing the Derivative of a Function $f ^ { \prime } ( x ) = \left( 4 x ^ { 4 } - 3 x ^ { 3 } + 6 x - 5 \right) ( 6 x ) + ( x - 3 ) \left( 6 x ^ { 2 } - 6 \right)$