Question

hw1q differentiation rules 1 (targets c1, c5; §3.3)
score: 2/8 answered: 2/8
question 3
if ( f(x)=6+\frac{4}{x}+\frac{7}{x^{2}} ), find ( f(x) ).

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Explanation:

Step1: Rewrite the function

Rewrite \(f(x)=6 + \frac{4}{x}+\frac{7}{x^{2}}\) as \(f(x)=6 + 4x^{-1}+7x^{-2}\).

Step2: Apply the power - rule for differentiation

The power - rule states that if \(y = ax^{n}\), then \(y^\prime=anx^{n - 1}\).
For the constant term \(6\), its derivative is \(0\) (since the derivative of a constant \(C\) is \(0\)).
For the term \(4x^{-1}\), using the power - rule: \((4x^{-1})^\prime=4\times(- 1)x^{-1 - 1}=-4x^{-2}\).
For the term \(7x^{-2}\), using the power - rule: \((7x^{-2})^\prime=7\times(-2)x^{-2 - 1}=-14x^{-3}\).

Step3: Combine the derivatives

\(f^\prime(x)=0-4x^{-2}-14x^{-3}=-\frac{4}{x^{2}}-\frac{14}{x^{3}}\)

Answer:

\(-\frac{4}{x^{2}}-\frac{14}{x^{3}}\)