Answer :
To find the derivative of [tex]\( y = (3x - 4)(4x^3 - x^2 + 1) \)[/tex], we can use the product rule. The product rule states that if you have a function [tex]\( y = u(x) \cdot v(x) \)[/tex], then its derivative [tex]\( y' \)[/tex] is given by:
[tex]\[
y' = u'v + uv'
\][/tex]
In this problem, let's identify the components:
- [tex]\( u(x) = 3x - 4 \)[/tex]
- [tex]\( v(x) = 4x^3 - x^2 + 1 \)[/tex]
First, we find the derivatives of [tex]\( u(x) \)[/tex] and [tex]\( v(x) \)[/tex]:
1. The derivative [tex]\( u'(x) \)[/tex] is:
[tex]\[
u'(x) = \frac{d}{dx}(3x - 4) = 3
\][/tex]
2. The derivative [tex]\( v'(x) \)[/tex] is:
[tex]\[
v'(x) = \frac{d}{dx}(4x^3 - x^2 + 1) = 12x^2 - 2x
\][/tex]
Now, apply the product rule:
[tex]\[
y' = u'v + uv'
\][/tex]
Substitute the derivatives and the original functions:
[tex]\[
y' = (3)(4x^3 - x^2 + 1) + (3x - 4)(12x^2 - 2x)
\][/tex]
Let's simplify each part individually:
1. [tex]\( u'v = 3(4x^3 - x^2 + 1) = 12x^3 - 3x^2 + 3 \)[/tex]
2. [tex]\( uv' = (3x - 4)(12x^2 - 2x) = (3x)(12x^2 - 2x) + (-4)(12x^2 - 2x) \)[/tex]
- Expand [tex]\( (3x)(12x^2 - 2x) = 36x^3 - 6x^2 \)[/tex]
- Expand [tex]\( (-4)(12x^2 - 2x) = -48x^2 + 8x \)[/tex]
Combine both terms:
[tex]\[
3x(12x^2 - 2x) + (-4)(12x^2 - 2x) = 36x^3 - 6x^2 - 48x^2 + 8x
\][/tex]
Simplify:
[tex]\[
= 36x^3 - 54x^2 + 8x
\][/tex]
Now combine [tex]\( u'v \)[/tex] and [tex]\( uv' \)[/tex]:
[tex]\[
y' = (12x^3 - 3x^2 + 3) + (36x^3 - 54x^2 + 8x)
\][/tex]
Combine like terms:
- Coefficients of [tex]\( x^3 \)[/tex]: [tex]\( 12x^3 + 36x^3 = 48x^3 \)[/tex]
- Coefficients of [tex]\( x^2 \)[/tex]: [tex]\( -3x^2 - 54x^2 = -57x^2 \)[/tex]
- Coefficients of [tex]\( x \)[/tex]: [tex]\( 8x \)[/tex]
- Constant term: [tex]\( + 3 \)[/tex]
Thus, the derivative is:
[tex]\[
y' = 48x^3 - 57x^2 + 8x + 3
\][/tex]
[tex]\[
y' = u'v + uv'
\][/tex]
In this problem, let's identify the components:
- [tex]\( u(x) = 3x - 4 \)[/tex]
- [tex]\( v(x) = 4x^3 - x^2 + 1 \)[/tex]
First, we find the derivatives of [tex]\( u(x) \)[/tex] and [tex]\( v(x) \)[/tex]:
1. The derivative [tex]\( u'(x) \)[/tex] is:
[tex]\[
u'(x) = \frac{d}{dx}(3x - 4) = 3
\][/tex]
2. The derivative [tex]\( v'(x) \)[/tex] is:
[tex]\[
v'(x) = \frac{d}{dx}(4x^3 - x^2 + 1) = 12x^2 - 2x
\][/tex]
Now, apply the product rule:
[tex]\[
y' = u'v + uv'
\][/tex]
Substitute the derivatives and the original functions:
[tex]\[
y' = (3)(4x^3 - x^2 + 1) + (3x - 4)(12x^2 - 2x)
\][/tex]
Let's simplify each part individually:
1. [tex]\( u'v = 3(4x^3 - x^2 + 1) = 12x^3 - 3x^2 + 3 \)[/tex]
2. [tex]\( uv' = (3x - 4)(12x^2 - 2x) = (3x)(12x^2 - 2x) + (-4)(12x^2 - 2x) \)[/tex]
- Expand [tex]\( (3x)(12x^2 - 2x) = 36x^3 - 6x^2 \)[/tex]
- Expand [tex]\( (-4)(12x^2 - 2x) = -48x^2 + 8x \)[/tex]
Combine both terms:
[tex]\[
3x(12x^2 - 2x) + (-4)(12x^2 - 2x) = 36x^3 - 6x^2 - 48x^2 + 8x
\][/tex]
Simplify:
[tex]\[
= 36x^3 - 54x^2 + 8x
\][/tex]
Now combine [tex]\( u'v \)[/tex] and [tex]\( uv' \)[/tex]:
[tex]\[
y' = (12x^3 - 3x^2 + 3) + (36x^3 - 54x^2 + 8x)
\][/tex]
Combine like terms:
- Coefficients of [tex]\( x^3 \)[/tex]: [tex]\( 12x^3 + 36x^3 = 48x^3 \)[/tex]
- Coefficients of [tex]\( x^2 \)[/tex]: [tex]\( -3x^2 - 54x^2 = -57x^2 \)[/tex]
- Coefficients of [tex]\( x \)[/tex]: [tex]\( 8x \)[/tex]
- Constant term: [tex]\( + 3 \)[/tex]
Thus, the derivative is:
[tex]\[
y' = 48x^3 - 57x^2 + 8x + 3
\][/tex]
