Answer :
To find the derivative of the function [tex]\( y = (x^2 - 5x + 2)(5x^3 - x^2 + 5) \)[/tex], we'll use the product rule for differentiation. The product rule states that if you have two functions multiplied together, say [tex]\( u(x) \)[/tex] and [tex]\( v(x) \)[/tex], then the derivative [tex]\( y' \)[/tex] is given by:
[tex]\[
y' = u'v + uv'
\][/tex]
Let's apply this to our function.
1. Identify the two functions:
- [tex]\( u(x) = x^2 - 5x + 2 \)[/tex]
- [tex]\( v(x) = 5x^3 - x^2 + 5 \)[/tex]
2. Compute the derivatives of [tex]\( u \)[/tex] and [tex]\( v \)[/tex]:
- The derivative of [tex]\( u(x) = x^2 - 5x + 2 \)[/tex] is:
[tex]\[
u'(x) = 2x - 5
\][/tex]
- The derivative of [tex]\( v(x) = 5x^3 - x^2 + 5 \)[/tex] is:
[tex]\[
v'(x) = 15x^2 - 2x
\][/tex]
3. Apply the product rule:
[tex]\[
y' = (2x - 5)(5x^3 - x^2 + 5) + (x^2 - 5x + 2)(15x^2 - 2x)
\][/tex]
4. Expand both terms:
- First term: [tex]\((2x - 5)(5x^3 - x^2 + 5)\)[/tex]
- Distribute [tex]\( 2x \)[/tex]:
[tex]\[
2x \cdot 5x^3 = 10x^4,\quad 2x \cdot (-x^2) = -2x^3,\quad 2x \cdot 5 = 10x
\][/tex]
- Distribute [tex]\(-5\)[/tex]:
[tex]\[
-5 \cdot 5x^3 = -25x^3,\quad -5 \cdot (-x^2) = 5x^2,\quad -5 \cdot 5 = -25
\][/tex]
- Combine:
[tex]\[
10x^4 - 2x^3 + 10x - 25x^3 + 5x^2 - 25 = 10x^4 - 27x^3 + 5x^2 + 10x - 25
\][/tex]
- Second term: [tex]\((x^2 - 5x + 2)(15x^2 - 2x)\)[/tex]
- Distribute [tex]\( x^2 \)[/tex]:
[tex]\[
x^2 \cdot 15x^2 = 15x^4,\quad x^2 \cdot (-2x) = -2x^3
\][/tex]
- Distribute [tex]\(-5x\)[/tex]:
[tex]\[
-5x \cdot 15x^2 = -75x^3,\quad -5x \cdot (-2x) = 10x^2
\][/tex]
- Distribute [tex]\( 2 \)[/tex]:
[tex]\[
2 \cdot 15x^2 = 30x^2,\quad 2 \cdot (-2x) = -4x
\][/tex]
- Combine:
[tex]\[
15x^4 - 2x^3 - 75x^3 + 10x^2 + 30x^2 - 4x = 15x^4 - 77x^3 + 40x^2 - 4x
\][/tex]
5. Combine all results:
[tex]\[
y' = (10x^4 - 27x^3 + 5x^2 + 10x - 25) + (15x^4 - 77x^3 + 40x^2 - 4x)
\][/tex]
6. Add the like terms:
- Combine [tex]\( x^4 \)[/tex]: [tex]\( 10x^4 + 15x^4 = 25x^4 \)[/tex]
- Combine [tex]\( x^3 \)[/tex]: [tex]\( -27x^3 - 77x^3 = -104x^3 \)[/tex]
- Combine [tex]\( x^2 \)[/tex]: [tex]\( 5x^2 + 40x^2 = 45x^2 \)[/tex]
- Combine [tex]\( x \)[/tex]: [tex]\( 10x - 4x = 6x \)[/tex]
- Constant: [tex]\(-25\)[/tex]
Therefore, the derivative is:
[tex]\[
y' = 25x^4 - 104x^3 + 45x^2 + 6x - 25
\][/tex]
So the correct answer is B. [tex]\( 25x^4 - 104x^3 + 45x^2 + 6x - 25 \)[/tex].
[tex]\[
y' = u'v + uv'
\][/tex]
Let's apply this to our function.
1. Identify the two functions:
- [tex]\( u(x) = x^2 - 5x + 2 \)[/tex]
- [tex]\( v(x) = 5x^3 - x^2 + 5 \)[/tex]
2. Compute the derivatives of [tex]\( u \)[/tex] and [tex]\( v \)[/tex]:
- The derivative of [tex]\( u(x) = x^2 - 5x + 2 \)[/tex] is:
[tex]\[
u'(x) = 2x - 5
\][/tex]
- The derivative of [tex]\( v(x) = 5x^3 - x^2 + 5 \)[/tex] is:
[tex]\[
v'(x) = 15x^2 - 2x
\][/tex]
3. Apply the product rule:
[tex]\[
y' = (2x - 5)(5x^3 - x^2 + 5) + (x^2 - 5x + 2)(15x^2 - 2x)
\][/tex]
4. Expand both terms:
- First term: [tex]\((2x - 5)(5x^3 - x^2 + 5)\)[/tex]
- Distribute [tex]\( 2x \)[/tex]:
[tex]\[
2x \cdot 5x^3 = 10x^4,\quad 2x \cdot (-x^2) = -2x^3,\quad 2x \cdot 5 = 10x
\][/tex]
- Distribute [tex]\(-5\)[/tex]:
[tex]\[
-5 \cdot 5x^3 = -25x^3,\quad -5 \cdot (-x^2) = 5x^2,\quad -5 \cdot 5 = -25
\][/tex]
- Combine:
[tex]\[
10x^4 - 2x^3 + 10x - 25x^3 + 5x^2 - 25 = 10x^4 - 27x^3 + 5x^2 + 10x - 25
\][/tex]
- Second term: [tex]\((x^2 - 5x + 2)(15x^2 - 2x)\)[/tex]
- Distribute [tex]\( x^2 \)[/tex]:
[tex]\[
x^2 \cdot 15x^2 = 15x^4,\quad x^2 \cdot (-2x) = -2x^3
\][/tex]
- Distribute [tex]\(-5x\)[/tex]:
[tex]\[
-5x \cdot 15x^2 = -75x^3,\quad -5x \cdot (-2x) = 10x^2
\][/tex]
- Distribute [tex]\( 2 \)[/tex]:
[tex]\[
2 \cdot 15x^2 = 30x^2,\quad 2 \cdot (-2x) = -4x
\][/tex]
- Combine:
[tex]\[
15x^4 - 2x^3 - 75x^3 + 10x^2 + 30x^2 - 4x = 15x^4 - 77x^3 + 40x^2 - 4x
\][/tex]
5. Combine all results:
[tex]\[
y' = (10x^4 - 27x^3 + 5x^2 + 10x - 25) + (15x^4 - 77x^3 + 40x^2 - 4x)
\][/tex]
6. Add the like terms:
- Combine [tex]\( x^4 \)[/tex]: [tex]\( 10x^4 + 15x^4 = 25x^4 \)[/tex]
- Combine [tex]\( x^3 \)[/tex]: [tex]\( -27x^3 - 77x^3 = -104x^3 \)[/tex]
- Combine [tex]\( x^2 \)[/tex]: [tex]\( 5x^2 + 40x^2 = 45x^2 \)[/tex]
- Combine [tex]\( x \)[/tex]: [tex]\( 10x - 4x = 6x \)[/tex]
- Constant: [tex]\(-25\)[/tex]
Therefore, the derivative is:
[tex]\[
y' = 25x^4 - 104x^3 + 45x^2 + 6x - 25
\][/tex]
So the correct answer is B. [tex]\( 25x^4 - 104x^3 + 45x^2 + 6x - 25 \)[/tex].
