Answer :
Final answer:
The derivative of the function f(x) = 9x^3 - x^2 + 6x - 7 is f'(x) = 27x^2 - 2x + 6, corresponding to option A).
Explanation:
To find the derivative of f(x), we apply the power rule and sum rule of differentiation. The power rule states that the derivative of x^n with respect to x is nx^(n-1). Applying this rule to each term of f(x), we obtain f'(x) = 27x^2 - 2x + 6.
For example, the derivative of 9x^3 is 27x^2, the derivative of -x^2 is -2x, and the derivative of 6x is 6. The derivative of a constant term, such as -7, is zero.
Therefore, the derivative of the function f(x) = 9x^3 - x^2 + 6x - 7 with respect to x is f'(x) = 27x^2 - 2x + 6, corresponding to option A). This result represents the rate of change of the function f(x) at any given point x and provides valuable information about its behavior and slope at that point.
So, the correct answer is A) 27x^2-2x+6
