Find $\frac{dy}{dx}$, if y = $\frac{2}{3}$ x$^3$ - $\frac{4}{x}$

Analysis

The goal is to find the derivative of the function $y = \frac{2}{3}x^3 - \frac{4}{x}$ with respect to $x$. This requires applying the Power Rule of differentiation.

Concept

The Power Rule states that if $y = ax^n$, then the derivative $\frac{dy}{dx} = n \cdot ax^{n-1}$.

Before differentiating, it is helpful to rewrite terms with $x$ in the denominator using negative exponents:

• $\frac{1}{x^n} = x^{-n}$

Step-by-Step Solution

Step 1: Rewrite the function The function is $y = \frac{2}{3}x^3 - \frac{4}{x}$. Rewrite the second term: $y = \frac{2}{3}x^3 - 4x^{-1}$

Step 2: Differentiate each term

• For the first term, $\frac{2}{3}x^3$: Multiply the coefficient by the power and subtract 1 from the power: $3 \times \frac{2}{3}x^{(3-1)} = 2x^2$

• For the second term, $-4x^{-1}$: Multiply the coefficient by the power and subtract 1 from the power: $-1 \times -4x^{(-1-1)} = 4x^{-2}$

Step 3: Combine the results $\frac{dy}{dx} = 2x^2 + 4x^{-2}$

Step 4: Convert back to fractional form Since $x^{-2} = \frac{1}{x^2}$, the expression becomes: $\frac{dy}{dx} = 2x^2 + \frac{4}{x^2}$

Conclusion

Comparing our result to the given options, we see that it matches Option B.

Exam Tip

Always remember to rewrite terms like $\frac{1}{x^n}$ as $x^{-n}$ before differentiating. A common mistake students make is trying to use the quotient rule unnecessarily; converting to a negative exponent makes the differentiation much faster and reduces the chance of errors!