Question

2.5-1
score: 6/9 answered: 6/9
question 7
if $f(x)=\frac{5x^{5}+2x^{4}-7x^{3}}{x^{4}}$, find $f(x)$
question help: video

Explanation:

Step1: Simplify the function

First, simplify $f(x)=\frac{5x^{5}+2x^{4}-7x^{3}}{x^{4}}$. Using the rule of dividing powers with the same base $\frac{a^{m}}{a^{n}}=a^{m - n}$, we get $f(x)=5x + 2-\frac{7}{x}=5x + 2-7x^{-1}$.

Step2: Apply the power - rule for differentiation

The power - rule states that if $y = ax^{n}$, then $y^\prime=anx^{n - 1}$, and the derivative of a constant is 0.
For the term $5x$, its derivative is $5\times1\times x^{1 - 1}=5$.
For the constant term 2, its derivative is 0.
For the term $-7x^{-1}$, its derivative is $-7\times(-1)\times x^{-1 - 1}=7x^{-2}=\frac{7}{x^{2}}$.

Step3: Combine the derivatives

$f^\prime(x)=5 + 0+\frac{7}{x^{2}}=5+\frac{7}{x^{2}}$.

Answer:

$f^\prime(x)=5+\frac{7}{x^{2}}$