Answer :
To find the derivative of the algebraic function [tex]\(H(x)=(x^5-3)(x^3+3)\)[/tex], we can use the product rule. The product rule states that if you have two functions [tex]\(u(x)\)[/tex] and [tex]\(v(x)\)[/tex], the derivative of their product is:
[tex]\[
(u \cdot v)' = u' \cdot v + u \cdot v'
\][/tex]
In this case, let:
- [tex]\(u(x) = x^5 - 3\)[/tex]
- [tex]\(v(x) = x^3 + 3\)[/tex]
First, let's find the derivatives of [tex]\(u(x)\)[/tex] and [tex]\(v(x)\)[/tex]:
1. [tex]\(u'(x) = \frac{d}{dx}(x^5 - 3) = 5x^4\)[/tex]
2. [tex]\(v'(x) = \frac{d}{dx}(x^3 + 3) = 3x^2\)[/tex]
Now, apply the product rule:
[tex]\[
H'(x) = u'(x) \cdot v(x) + u(x) \cdot v'(x)
\][/tex]
Substitute the derivatives and original functions into the product rule:
[tex]\[
H'(x) = (5x^4) \cdot (x^3 + 3) + (x^5 - 3) \cdot (3x^2)
\][/tex]
Expand both products:
1. [tex]\(5x^4 \cdot (x^3 + 3) = 5x^4 \cdot x^3 + 5x^4 \cdot 3 = 5x^7 + 15x^4\)[/tex]
2. [tex]\((x^5 - 3) \cdot 3x^2 = x^5 \cdot 3x^2 - 3 \cdot 3x^2 = 3x^7 - 9x^2\)[/tex]
Combine the results:
[tex]\[
H'(x) = 5x^7 + 15x^4 + 3x^7 - 9x^2
\][/tex]
Combine like terms:
[tex]\[
H'(x) = (5x^7 + 3x^7) + 15x^4 - 9x^2 = 8x^7 + 15x^4 - 9x^2
\][/tex]
Thus, the derivative of [tex]\(H(x)\)[/tex] is:
[tex]\[
H'(x) = 8x^7 + 15x^4 - 9x^2
\][/tex]
So, the correct answer is:
[tex]\[
H'(x) = 8x^7 + 15x^4 - 9x^2
\][/tex]
This matches the option [tex]\(H^{\prime}(s)=8 x^7+15 x^4-9 x^2\)[/tex].
[tex]\[
(u \cdot v)' = u' \cdot v + u \cdot v'
\][/tex]
In this case, let:
- [tex]\(u(x) = x^5 - 3\)[/tex]
- [tex]\(v(x) = x^3 + 3\)[/tex]
First, let's find the derivatives of [tex]\(u(x)\)[/tex] and [tex]\(v(x)\)[/tex]:
1. [tex]\(u'(x) = \frac{d}{dx}(x^5 - 3) = 5x^4\)[/tex]
2. [tex]\(v'(x) = \frac{d}{dx}(x^3 + 3) = 3x^2\)[/tex]
Now, apply the product rule:
[tex]\[
H'(x) = u'(x) \cdot v(x) + u(x) \cdot v'(x)
\][/tex]
Substitute the derivatives and original functions into the product rule:
[tex]\[
H'(x) = (5x^4) \cdot (x^3 + 3) + (x^5 - 3) \cdot (3x^2)
\][/tex]
Expand both products:
1. [tex]\(5x^4 \cdot (x^3 + 3) = 5x^4 \cdot x^3 + 5x^4 \cdot 3 = 5x^7 + 15x^4\)[/tex]
2. [tex]\((x^5 - 3) \cdot 3x^2 = x^5 \cdot 3x^2 - 3 \cdot 3x^2 = 3x^7 - 9x^2\)[/tex]
Combine the results:
[tex]\[
H'(x) = 5x^7 + 15x^4 + 3x^7 - 9x^2
\][/tex]
Combine like terms:
[tex]\[
H'(x) = (5x^7 + 3x^7) + 15x^4 - 9x^2 = 8x^7 + 15x^4 - 9x^2
\][/tex]
Thus, the derivative of [tex]\(H(x)\)[/tex] is:
[tex]\[
H'(x) = 8x^7 + 15x^4 - 9x^2
\][/tex]
So, the correct answer is:
[tex]\[
H'(x) = 8x^7 + 15x^4 - 9x^2
\][/tex]
This matches the option [tex]\(H^{\prime}(s)=8 x^7+15 x^4-9 x^2\)[/tex].
