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Answer :
To find the derivative of the function [tex]\( f(x) = 6x^4 + 5x^3 - 9 \)[/tex], we will use basic rules of differentiation. Here's a step-by-step explanation:
1. Identify the function components: The function [tex]\( f(x) = 6x^4 + 5x^3 - 9 \)[/tex] is a polynomial, and we need to find the derivative with respect to [tex]\( x \)[/tex].
2. Apply the power rule: The power rule states that the derivative of [tex]\( x^n \)[/tex] is [tex]\( n \cdot x^{n-1} \)[/tex]. We will use this rule to differentiate each term of the polynomial individually.
3. Differentiate [tex]\( 6x^4 \)[/tex]:
- Using the power rule, the derivative of [tex]\( 6x^4 \)[/tex] is [tex]\( 4 \cdot 6x^{4-1} = 24x^3 \)[/tex].
4. Differentiate [tex]\( 5x^3 \)[/tex]:
- Again, applying the power rule, the derivative of [tex]\( 5x^3 \)[/tex] is [tex]\( 3 \cdot 5x^{3-1} = 15x^2 \)[/tex].
5. Differentiate the constant [tex]\(-9\)[/tex]:
- The derivative of a constant is [tex]\( 0 \)[/tex].
6. Combine the results:
- Adding up the derivatives of each term, we get the derivative of the entire function:
[tex]\[
\frac{d}{dx}(6x^4 + 5x^3 - 9) = 24x^3 + 15x^2 + 0
\][/tex]
Simplifying this, we have:
[tex]\[
\frac{d}{dx}(6x^4 + 5x^3 - 9) = 24x^3 + 15x^2
\][/tex]
Therefore, the derivative of the function is [tex]\( 24x^3 + 15x^2 \)[/tex].
1. Identify the function components: The function [tex]\( f(x) = 6x^4 + 5x^3 - 9 \)[/tex] is a polynomial, and we need to find the derivative with respect to [tex]\( x \)[/tex].
2. Apply the power rule: The power rule states that the derivative of [tex]\( x^n \)[/tex] is [tex]\( n \cdot x^{n-1} \)[/tex]. We will use this rule to differentiate each term of the polynomial individually.
3. Differentiate [tex]\( 6x^4 \)[/tex]:
- Using the power rule, the derivative of [tex]\( 6x^4 \)[/tex] is [tex]\( 4 \cdot 6x^{4-1} = 24x^3 \)[/tex].
4. Differentiate [tex]\( 5x^3 \)[/tex]:
- Again, applying the power rule, the derivative of [tex]\( 5x^3 \)[/tex] is [tex]\( 3 \cdot 5x^{3-1} = 15x^2 \)[/tex].
5. Differentiate the constant [tex]\(-9\)[/tex]:
- The derivative of a constant is [tex]\( 0 \)[/tex].
6. Combine the results:
- Adding up the derivatives of each term, we get the derivative of the entire function:
[tex]\[
\frac{d}{dx}(6x^4 + 5x^3 - 9) = 24x^3 + 15x^2 + 0
\][/tex]
Simplifying this, we have:
[tex]\[
\frac{d}{dx}(6x^4 + 5x^3 - 9) = 24x^3 + 15x^2
\][/tex]
Therefore, the derivative of the function is [tex]\( 24x^3 + 15x^2 \)[/tex].
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