Question

Use the definition of the derivative to find f'(x).

Answer 1

The definition of a derivative is the limit of the difference quotient [f(x + h) - f(x)] / h as h approaches 0. Without a specific function provided, we illustrated this with the example function f(x) = x², yielding the derivative f'(x) = 2x.

Step-by-step explanation:

To find the derivative of a function using the definition of the derivative, we need to understand the limit process. The definition is given by:

f'(x) = lim(h → 0) [f(x + h) - f(x)] / h

We need a specific function f(x) to apply this process, which is not provided in the student's question. However, here is a general step-by-step method:

1. Identify the function f(x).
2. Calculate f(x + h).
3. Form the difference quotient [f(x + h) - f(x)] / h.
4. Take the limit as h approaches 0.

For instance, if f(x) = x², then f(x + h) = (x + h)². The difference quotient is [(x + h)² - x²] / h, which simplifies to (2xh + h²) / h. Taking the limit as h approaches 0, we find that f'(x) = 2x.

Remember that the derivative f'(x) represents the instantaneous rate of change of the function f(x) at any point x.