Question

the derivative of f(x) = 9x³ - 6x⁴ is f(x) = bx^k. find the value of b. give an exact answer either as a decimal or as a fraction (in the form n/m) in fully simplified form.

Explanation:

Step1: Rewrite the function

Rewrite \( f(x)=\frac{9}{x^{3}}-6x^{4} \) as \( f(x) = 9x^{-3}-6x^{4} \).

Step2: Apply the power rule

The power rule is \( \frac{d}{dx}(x^{n})=nx^{n - 1} \).
For the first term \( y = 9x^{-3} \), using the power rule: \( \frac{d}{dx}(9x^{-3})=9\times(-3)x^{-3 - 1}=-27x^{-4} \).
For the second term \( y=-6x^{4} \), using the power rule: \( \frac{d}{dx}(-6x^{4})=-6\times4x^{4 - 1}=-24x^{3} \).

Answer:

\( f^{\prime}(x)=-27x^{-4}-24x^{3} \), so \( b=-27 \) and \( k = - 4 \) (if we consider the form \( f^{\prime}(x)=bx^{k}\) for the first - term \( -27x^{-4}\)).