Final Answer:
The f'(x) is equal to is 2x (option b).
Step-by-step explanation:
To find the derivative f'(x) of the function f(x) = x^2 using the given expression f(x + h) - f(x) = h, we employ the difference quotient formula. The difference quotient is given by (f(x + h) - f(x))/h. Substituting f(x) = x^2 into this formula, we get ((x + h)^2 - x^2)/h. Simplifying this expression leads to (2xh + h^2)/h, and canceling h results in 2x + h. As h approaches 0, the derivative f'(x) becomes 2x (option b).
The calculation involves understanding the concept of the derivative as the limit of the difference quotient as h approaches 0. By applying the difference quotient formula to the given function f(x) = x^2, we obtain an expression that simplifies to 2x + h. As h tends to 0, the h term becomes negligible, leaving 2x as the derivative f'(x). Therefore, the correct answer is b. 2x.
In conclusion, the derivative f'(x) of the function f(x) = x^2 with respect to x is 2x, as obtained through the application of the difference quotient formula and the limit as h approaches 0. This mathematical process illustrates the fundamental concept of derivatives in calculus, providing a clear understanding of how the rate of change of the function is calculated.