If y = 4x3 - 2x2 + x, find $\frac{\delta y}{\delta x}$

Analysis

The question asks for the derivative of the function $y = 4x^3 - 2x^2 + x$ 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:

In simple terms:

1. Bring the exponent down to multiply with the coefficient.
2. Subtract 1 from the original exponent.
3. Remember that the derivative of $x$ (which is $x^1$) is $1$, and the derivative of a constant is $0$.

Step-by-Step Solution

Given the function:

Step 1: Differentiate $4x^3$ Multiply the exponent (3) by the coefficient (4) and subtract 1 from the exponent:

Step 2: Differentiate $-2x^2$ Multiply the exponent (2) by the coefficient (-2) and subtract 1 from the exponent:

Step 3: Differentiate $x$ The derivative of $x$ is simply 1:

Step 4: Combine the results

Conclusion

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

Exam Tip

When differentiating polynomials, always check the signs carefully. A common mistake is to forget to subtract 1 from the exponent or to miscalculate the product of the exponent and the coefficient. Also, remember that the derivative of any constant (like 5 or 10) is always 0, but the derivative of $x$ is always 1.