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Answer :
To find the derivative of the function [tex]\( f(x) = x^5 - 3x^3 \)[/tex], we'll use basic differentiation rules. Here's a step-by-step explanation:
1. Identify the terms in the function:
- [tex]\( x^5 \)[/tex]
- [tex]\(-3x^3\)[/tex]
2. Differentiate each term separately:
- The derivative of [tex]\( x^5 \)[/tex] with respect to [tex]\( x \)[/tex] is found using the power rule, which states that the derivative of [tex]\( x^n \)[/tex] is [tex]\( nx^{n-1} \)[/tex]. Here, [tex]\( n = 5 \)[/tex], so its derivative is:
[tex]\[
5x^{5-1} = 5x^4
\][/tex]
- For the term [tex]\(-3x^3\)[/tex], apply the power rule similarly. The constant coefficient [tex]\(-3\)[/tex] is carried along:
[tex]\[
-3 \times 3x^{3-1} = -9x^2
\][/tex]
3. Combine the derivatives:
- Add the derivatives of the individual terms to obtain the derivative of the entire function:
[tex]\[
5x^4 - 9x^2
\][/tex]
Thus, the derivative of the function [tex]\( f(x) = x^5 - 3x^3 \)[/tex] is [tex]\( 5x^4 - 9x^2 \)[/tex].
Given the multiple-choice options, this matches option (a): [tex]\( 5x^4 - 9x^2 \)[/tex].
1. Identify the terms in the function:
- [tex]\( x^5 \)[/tex]
- [tex]\(-3x^3\)[/tex]
2. Differentiate each term separately:
- The derivative of [tex]\( x^5 \)[/tex] with respect to [tex]\( x \)[/tex] is found using the power rule, which states that the derivative of [tex]\( x^n \)[/tex] is [tex]\( nx^{n-1} \)[/tex]. Here, [tex]\( n = 5 \)[/tex], so its derivative is:
[tex]\[
5x^{5-1} = 5x^4
\][/tex]
- For the term [tex]\(-3x^3\)[/tex], apply the power rule similarly. The constant coefficient [tex]\(-3\)[/tex] is carried along:
[tex]\[
-3 \times 3x^{3-1} = -9x^2
\][/tex]
3. Combine the derivatives:
- Add the derivatives of the individual terms to obtain the derivative of the entire function:
[tex]\[
5x^4 - 9x^2
\][/tex]
Thus, the derivative of the function [tex]\( f(x) = x^5 - 3x^3 \)[/tex] is [tex]\( 5x^4 - 9x^2 \)[/tex].
Given the multiple-choice options, this matches option (a): [tex]\( 5x^4 - 9x^2 \)[/tex].
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