The correct answer is A.
The derivative of a function is a measure of how much the function changes as its input changes. Intuitively, the derivative of a function at a point tells us how much the function “goes up” or “goes down” at that point.
The derivative of a symmetric function is always zero. This is because a symmetric function is a function that is unchanged when its input is reflected across the y-axis. In other words, if $f(x)$ is a symmetric function, then $f(-x) = f(x)$.
The derivative of a function can be calculated using the limit definition of the derivative:
$$f'(x) = \lim_{h \to 0} \frac{f(x+h) – f(x)}{h}$$
In the case of a symmetric function, we can use the symmetry of the function to simplify the limit:
$$f'(x) = \lim_{h \to 0} \frac{f(x+h) – f(x)}{h} = \lim_{h \to 0} \frac{f(-x-h) – f(-x)}{h} = \lim_{h \to 0} \frac{f(-x) – f(-x)}{h} = 0$$
Therefore, the derivative of a symmetric function is always zero.
Option A is the only option that is a horizontal line. A horizontal line has a slope of zero, which is the derivative of a symmetric function.