the derivative of f(x) = 9x³ - 6x⁴ is f(x) = bx^k. find the value of b. give an exact answer either as a…

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.

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.

Answer

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})).