If y = 2x3 + 6x2 + 6x + 1, Find $\frac{dy}{dx}$

Analysis

The question asks for the derivative of the polynomial function $y = 2x^3 + 6x^2 + 6x + 1$ with respect to $x$. This is a standard calculus problem involving the power rule for differentiation.

Concept: The Power Rule

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

Essentially, you multiply the coefficient by the exponent and then subtract 1 from the exponent.

• The derivative of a constant (like 1) is always 0.
• The derivative of $x$ (which is $x^1$) is 1.

Step-by-Step Solution

Given the function:

We differentiate each term individually:

1. For $2x^3$: Multiply 3 by 2 and subtract 1 from the exponent: $3 \times 2x^{(3-1)} = 6x^2$

2. For $6x^2$: Multiply 2 by 6 and subtract 1 from the exponent: $2 \times 6x^{(2-1)} = 12x^1 = 12x$

3. For $6x$: The derivative of $6x$ is simply the coefficient: $1 \times 6x^{(1-1)} = 6x^0 = 6(1) = 6$

4. For $1$: The derivative of any constant is 0.

Combining these results:

Conclusion

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

Exam Tip: Always remember to differentiate the constant term! Many students mistakenly leave the constant in the final answer or forget to change it to zero. Also, double-check your arithmetic when multiplying the exponent by the coefficient—it's a common place for "silly" mistakes under exam pressure.