Derivative of the Function \( f(x) = 2 + \frac{8}{x} + \frac{6}{x^2} \) Using Power Rule
Step 1: Rewrite the Function Using Negative Exponents
\[ f(x) = 2 + 8x^{-1} + 6x^{-2} \]
Step 2: Apply the Power Rule to Each Term
The Power Rule states that if \( f(x) = ax^n \), then \( f'(x) = n \cdot ax^{(n-1)} \).
Derivative of the First Term
The derivative of the constant term \( 2 \) is \( 0 \).
Derivative of the Second Term
\[ \frac{d}{dx}(8x^{-1}) = -1 \cdot 8x^{(-1 – 1)} = -8x^{-2} \]
Derivative of the Third Term
\[ \frac{d}{dx}(6x^{-2}) = -2 \cdot 6x^{(-2 – 1)} = -12x^{-3} \]
Step 3: Combine the Derivatives
\[ f'(x) = 0 + (-8x^{-2}) + (-12x^{-3}) = -8x^{-2} – 12x^{-3} \]
Step 4: Rewrite the Derivative Without Negative Exponents
\[ f'(x) = -8\left(\frac{1}{x^2}\right) – 12\left(\frac{1}{x^3}\right) \]
Step 5: Rewrite the Derivative with Constants on Top
\[ f'(x) = -\frac{8}{x^2} – \frac{12}{x^3} \]