If y = 8x$^3$ - 3x$^2$ + 7x - 1, find $\frac{\mathrm d^2 y}{\mathrm d x^2}$.

Analysis

The question asks for the second derivative ($\frac{\mathrm d^2 y}{\mathrm d x^2}$) of the given polynomial function $y = 8x^3 - 3x^2 + 7x - 1$. To find the second derivative, we must differentiate the function with respect to $x$ twice.

Concept: The Power Rule

To differentiate a term of the form $ax^n$, we use the Power Rule:

The derivative of a constant (a number without $x$) is always $0$.

Step-by-Step Solution

Step 1: Find the first derivative ($\frac{\mathrm d y}{\mathrm d x}$) We differentiate each term of $y = 8x^3 - 3x^2 + 7x - 1$ individually:

• For $8x^3$: $3 \times 8x^{3-1} = 24x^2$
• For $-3x^2$: $2 \times -3x^{2-1} = -6x$
• For $7x$: $1 \times 7x^{1-1} = 7x^0 = 7$
• For $-1$: The derivative is $0$

So, the first derivative is:

Step 2: Find the second derivative ($\frac{\mathrm d^2 y}{\mathrm d x^2}$) Now, we differentiate the result from Step 1:

• For $24x^2$: $2 \times 24x^{2-1} = 48x$
• For $-6x$: $1 \times -6x^{1-1} = -6$
• For $7$: The derivative is $0$

Combining these, we get:

Conclusion

Comparing our result to the given options, we find that it matches option A.

Exam Tip: When dealing with polynomials, remember that every time you differentiate, the power of $x$ decreases by 1. If you are asked for the $n^{th}$ derivative, you simply repeat the process $n$ times. Always double-check your signs (positive/negative) during the multiplication step!