Question

hw10 differentiation rules 1 (targets c1, c5; §3.3)
score: 6/9 answered: 6/9
question 7
if (f(x)=\frac{-2x^{5}-7x^{4}-4x^{3}}{x^{4}}), find (f(x))
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Explanation:

Step1: Simplify the function

First, divide each term in the numerator by $x^{4}$:
$f(x)=\frac{-2x^{5}}{x^{4}}-\frac{7x^{4}}{x^{4}}-\frac{4x^{3}}{x^{4}}=-2x - 7-\frac{4}{x}=-2x - 7-4x^{-1}$

Step2: Differentiate term - by - term

Using the power rule $\frac{d}{dx}(ax^{n})=nax^{n - 1}$, for the first term: $\frac{d}{dx}(-2x)=-2$.
For the second term: $\frac{d}{dx}(-7) = 0$ (since the derivative of a constant is 0).
For the third term: $\frac{d}{dx}(-4x^{-1})=-4\times(-1)x^{-1 - 1}=4x^{-2}=\frac{4}{x^{2}}$.

Step3: Combine the derivatives

$f'(x)=-2 + 0+\frac{4}{x^{2}}=-2+\frac{4}{x^{2}}$

Answer:

$-2+\frac{4}{x^{2}}$