Question

2.5-1 score: 6/9 answered: 6/9 question 7 if $f(x)=\frac{-7x^{5}-4x^{4}-4x^{3}}{x^{4}}$, find $f(x)$ question help: video

Explanation:

Step1: Simplify the function

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

Step2: Apply the power - rule for differentiation

The power - rule states that if $y = ax^{n}$, then $y'=nax^{n - 1}$.
For the term $-7x$: The derivative of $-7x$ (where $a=-7$ and $n = 1$) is $-7\times1\times x^{1 - 1}=-7$.
For the constant term $-4$: The derivative of a constant is $0$.
For the term $-4x^{-1}$: The derivative of $-4x^{-1}$ (where $a=-4$ and $n=-1$) is $(-1)\times(-4)x^{-1 - 1}=4x^{-2}=\frac{4}{x^{2}}$.

Step3: Combine the derivatives of each term

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

Answer:

$-7+\frac{4}{x^{2}}$