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Answer :
Let's solve the problem step by step!
1. Given Polynomial:
The polynomial we are working with is:
[tex]\[
45c^5 - 63c^3 + 27c
\][/tex]
2. Finding the Derivative:
To find the derivative of the polynomial, we apply the power rule to each term. The power rule states that the derivative of [tex]\(c^n\)[/tex] is [tex]\(n \cdot c^{n-1}\)[/tex].
- Derivative of [tex]\(45c^5\)[/tex]:
[tex]\[
5 \times 45c^{5-1} = 225c^4
\][/tex]
- Derivative of [tex]\(-63c^3\)[/tex]:
[tex]\[
3 \times -63c^{3-1} = -189c^2
\][/tex]
- Derivative of [tex]\(27c\)[/tex]:
[tex]\[
1 \times 27c^{1-1} = 27
\][/tex]
Combining these, the derivative of the polynomial is:
[tex]\[
225c^4 - 189c^2 + 27
\][/tex]
3. Finding the Integral:
To find the integral, we apply the reverse of the power rule. The integral of [tex]\(c^n\)[/tex] is [tex]\(\frac{c^{n+1}}{n+1}\)[/tex].
- Integral of [tex]\(45c^5\)[/tex]:
[tex]\[
\frac{45c^{5+1}}{5+1} = \frac{45c^6}{6} = \frac{15c^6}{2}
\][/tex]
- Integral of [tex]\(-63c^3\)[/tex]:
[tex]\[
\frac{-63c^{3+1}}{3+1} = \frac{-63c^4}{4} = -\frac{63c^4}{4}
\][/tex]
- Integral of [tex]\(27c\)[/tex]:
[tex]\[
\frac{27c^{1+1}}{1+1} = \frac{27c^2}{2}
\][/tex]
Combining these, the integral of the polynomial is:
[tex]\[
\frac{15c^6}{2} - \frac{63c^4}{4} + \frac{27c^2}{2} + C
\][/tex]
where [tex]\(C\)[/tex] is the constant of integration.
In summary, for the polynomial [tex]\(45c^5 - 63c^3 + 27c\)[/tex]:
- Derivative: [tex]\(225c^4 - 189c^2 + 27\)[/tex]
- Integral: [tex]\(\frac{15c^6}{2} - \frac{63c^4}{4} + \frac{27c^2}{2} + C\)[/tex] where [tex]\(C\)[/tex] is the constant of integration.
1. Given Polynomial:
The polynomial we are working with is:
[tex]\[
45c^5 - 63c^3 + 27c
\][/tex]
2. Finding the Derivative:
To find the derivative of the polynomial, we apply the power rule to each term. The power rule states that the derivative of [tex]\(c^n\)[/tex] is [tex]\(n \cdot c^{n-1}\)[/tex].
- Derivative of [tex]\(45c^5\)[/tex]:
[tex]\[
5 \times 45c^{5-1} = 225c^4
\][/tex]
- Derivative of [tex]\(-63c^3\)[/tex]:
[tex]\[
3 \times -63c^{3-1} = -189c^2
\][/tex]
- Derivative of [tex]\(27c\)[/tex]:
[tex]\[
1 \times 27c^{1-1} = 27
\][/tex]
Combining these, the derivative of the polynomial is:
[tex]\[
225c^4 - 189c^2 + 27
\][/tex]
3. Finding the Integral:
To find the integral, we apply the reverse of the power rule. The integral of [tex]\(c^n\)[/tex] is [tex]\(\frac{c^{n+1}}{n+1}\)[/tex].
- Integral of [tex]\(45c^5\)[/tex]:
[tex]\[
\frac{45c^{5+1}}{5+1} = \frac{45c^6}{6} = \frac{15c^6}{2}
\][/tex]
- Integral of [tex]\(-63c^3\)[/tex]:
[tex]\[
\frac{-63c^{3+1}}{3+1} = \frac{-63c^4}{4} = -\frac{63c^4}{4}
\][/tex]
- Integral of [tex]\(27c\)[/tex]:
[tex]\[
\frac{27c^{1+1}}{1+1} = \frac{27c^2}{2}
\][/tex]
Combining these, the integral of the polynomial is:
[tex]\[
\frac{15c^6}{2} - \frac{63c^4}{4} + \frac{27c^2}{2} + C
\][/tex]
where [tex]\(C\)[/tex] is the constant of integration.
In summary, for the polynomial [tex]\(45c^5 - 63c^3 + 27c\)[/tex]:
- Derivative: [tex]\(225c^4 - 189c^2 + 27\)[/tex]
- Integral: [tex]\(\frac{15c^6}{2} - \frac{63c^4}{4} + \frac{27c^2}{2} + C\)[/tex] where [tex]\(C\)[/tex] is the constant of integration.
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