If $y = 6x^3 + 2x^{-2} - x^{-3}$, find $\frac{\mathrm d y}{\mathrm d x}$.

Analysis

To find $\frac{\mathrm{d}y}{\mathrm{d}x}$, we need to apply the Power Rule of Differentiation. The power rule states that if $y = ax^n$, then $\frac{\mathrm{d}y}{\mathrm{d}x} = n \cdot ax^{n-1}$.

Concept

The Power Rule is the fundamental tool for differentiating polynomials and terms with negative or fractional exponents. You simply:

1. Bring the current exponent down to multiply with the coefficient.
2. Subtract 1 from the current exponent.

Step-by-Step Solution

Given the function: $y = 6x^3 + 2x^{-2} - x^{-3}$

We differentiate each term individually:

1. For $6x^3$: • Bring down the 3: $3 \times 6 = 18$ • Subtract 1 from the exponent: $3 - 1 = 2$ • Result: $18x^2$

2. For $2x^{-2}$: • Bring down the -2: $-2 \times 2 = -4$ • Subtract 1 from the exponent: $-2 - 1 = -3$ • Result: $-4x^{-3}$

3. For $-x^{-3}$: • Bring down the -3: $-3 \times -1 = +3$ • Subtract 1 from the exponent: $-3 - 1 = -4$ • Result: $+3x^{-4}$

Combining these results, we get: $\frac{\mathrm{d}y}{\mathrm{d}x} = 18x^2 - 4x^{-3} + 3x^{-4}$

Conclusion

Comparing our result to the given options, it matches option C perfectly.

Exam Tip

When dealing with negative exponents, be very careful with your signs during subtraction. Many students mistakenly calculate $-2 - 1$ as $-1$ instead of $-3$. Always remember: "Subtracting 1 from a negative number makes it more negative."