if ( f(x)=\frac{-4 x^{5}+2 x^{4}-7 x^{3}}{x^{4}} ), find ( f^{prime}(x) ) hint: this one is easier if you…

Explanation:

Step1: Simplify the function

Divide each term in the numerator by (x^{4}): [ \begin{align*} f(x)&=\frac{-4x^{5}}{x^{4}}+\frac{2x^{4}}{x^{4}}-\frac{7x^{3}}{x^{4}}\ &=- 4x+2 - 7x^{-1} \end{align*} ]

Step2: Apply the power rule

The power rule is (\frac{d}{dx}(x^{n})=nx^{n - 1}). For (y=-4x), (\frac{d}{dx}(-4x)=-4) (since (n = 1), (1\times(-4)x^{1-1}=-4)). For (y = 2) (a constant), (\frac{d}{dx}(2)=0) (since (\frac{d}{dx}(c)=0) where (c) is a constant). For (y=-7x^{-1}), (\frac{d}{dx}(-7x^{-1})=-7\times(-1)x^{-1 - 1}=7x^{-2}=\frac{7}{x^{2}})

Step3: Combine the derivatives

(f^{\prime}(x)=-4+0+\frac{7}{x^{2}})

Answer:

(f^{\prime}(x)=-4+\frac{7}{x^{2}})