Question

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 divide each term by the common denominator before applying the power rule of derivatives. question help: video message instructor submit question jump to answer

Explanation:

Step1: Simplify the function

Divide each term in the numerator by \(x^{4}\):

$$
LATEXBLOCK0
$$

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:

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