College

Find [tex]\frac{d}{dx}\left(6x^4 + 5x^3 - 9\right)[/tex].

A. [tex]6x^3 + 5x^2 - 9[/tex]

B. [tex]24x^3 + 15x^2 - 9[/tex]

C. [tex]24x^3 + 15x^2[/tex]

D. [tex]6x^3 + 5x^2[/tex]

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].