Final answer:
Option A. To find the derivative of 2x² using the limit definition, we can apply the formula. The derivative of 2x² is 4x.
Step-by-step explanation:
To find the derivative of 2x² using the limit definition, we can apply the formula: lim(h→0) [f(x+h) - f(x)] / h. In this case, f(x) = 2x². Let's substitute this into the formula and simplify step-by-step:
- lim(h→0) [(2(x+h)²) - (2x²)] / h
- lim(h→0) [(2(x² + 2xh + h²)) - (2x²)] / h
- lim(h→0) [(2x² + 4xh + 2h²) - 2x²] / h
- lim(h→0) [4xh + 2h²] / h
- lim(h→0) 4x + 2h
Finally, as h approaches 0, the limit simplifies to 4x. Therefore, the derivative of 2x² is 4x.