To find the derivative of $q(x) = (0.5x)^2$, we will apply the power rule and simplify the expression.

Steps:

1. Expand the function: $q(x) = (0.5x)^2 = 0.25x^2$ (This is because $(0.5)^2 = 0.25$).

2. Differentiate using the power rule: The power rule for differentiation states that if $f(x) = ax^n$, then $f'(x) = n \cdot ax^{n-1}$.

   Here, $a = 0.25$ and $n = 2$, so: $q'(x) = 2 \cdot 0.25x^{2-1} = 0.5x$

Final Answer:

The derivative of $q(x) = (0.5x)^2$ is: $q'(x) = 0.5x$

Would you like any further clarification or have more questions about derivatives?

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Related Questions:

1. How would you differentiate $f(x) = (3x)^2$?
2. What is the derivative of $p(x) = 5x^3$?
3. Can you explain the power rule for derivatives in more detail?
4. How do you find the derivative of a function that has both multiplication and addition, like $g(x) = (2x + 3)^2$?
5. What is the second derivative of $q(x) = 0.25x^2$?

Tip:

Remember that when differentiating powers of $x$, always subtract 1 from the exponent and multiply by the original exponent.