Answer :

The first derivative of the function [tex]g(x) = 6x^3 + 45x^2 + 72x[/tex]is [tex]g'(x) = 18x^2 + 90x + 72[/tex], which is determined by applying the power rule and constant multiple rule of differentiation.

To find the first derivative, we apply the power rule and constant multiple rule of differentiation. The power rule states that if we have a term of the form[tex]x^n[/tex], the derivative is [tex]nx^(n-1)[/tex].

In this case, we have three terms: [tex]6x^3[/tex], [tex]45x^2[/tex], and 72x. Applying the power rule to each term, we get:

- The derivative of [tex]6x^3 is (3)(6)x^(3-1) = 18x^2[/tex].

- The derivative of [tex]45x^2 is (2)(45)x^(2-1) = 90x[/tex].

- The derivative of [tex]72x is (1)(72)x^(1-1) = 72[/tex].

Combining these derivatives, we obtain the first derivative of g(x):

[tex]g'(x) = 18x^2 + 90x + 72.[/tex]

This derivative represents the rate of change of the function g(x) with respect to x. It gives us information about the slope of the tangent line to the graph of g(x) at any point.

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